#348094

order	512
nilpotency class	3
isoclinism	isoclinic class of #348090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3529	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

845 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 845 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,f^c=fi,g^c=ghi,f^d=fhi,g^d=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255290);