#488924

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61086	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

962 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,7)(5,8)(9,10,16,22)(11,23,27,53)(12,18,30,45)(13,21,15,19)(14,33,37,47)(17,39,42,26)(20,32,50,36)(24,54,57,52)(25,59,38,48)(28,46,31,43)(29,62,35,65)(34,51,67,41)(40,60,69,56)(44,64,49,61)(55,71,58,72)(63,68,66,70);(2,6)(3,4)(7,8)(9,11,16,27)(10,17,22,42)(12,24,30,57)(13,25,15,38)(14,34,37,67)(18,40,45,69)(19,41,21,51)(20,48,50,59)(23,47,53,33)(26,32,39,36)(28,55,31,58)(29,63,35,66)(43,68,46,70)(44,71,49,72)(52,62,54,65)(56,64,60,61);(2,7)(3,4)(6,8)(9,12)(10,18)(11,24)(13,31)(14,35)(15,28)(16,30)(17,40)(19,46)(20,49)(21,43)(22,45)(23,52)(25,58)(26,60)(27,57)(29,37)(32,61)(33,65)(34,66)(36,64)(38,55)(39,56)(41,70)(42,69)(44,50)(47,62)(48,72)(51,68)(53,54)(59,71)(63,67);(9,13,16,15)(10,19,22,21)(11,25,27,38)(12,28,30,31)(14,32,37,36)(17,41,42,51)(18,43,45,46)(20,47,50,33)(23,48,53,59)(24,55,57,58)(26,34,39,67)(29,61,35,64)(40,68,69,70)(44,62,49,65)(52,71,54,72)(56,66,60,63)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396120);