#491512

order	512
nilpotency class	4
isoclinism	isoclinic class of #491511
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

611 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398708);