#6685127

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42806	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,19,26,35)(16,27,29,20)(17,21,31,38)(23,32,25,34)(24,36,28,33)(30,41,42,37)(39,44,40,43);(6,9)(7,11)(13,15,18,26)(14,19,22,35)(16,28,29,24)(17,23,31,25)(20,36,27,33)(21,32,38,34)(30,40,42,39)(37,44,41,43);(1,2,3,4)(15,23)(16,29)(17,31)(19,32)(20,27)(21,38)(24,40)(25,26)(28,39)(33,44)(34,35)(36,43);(6,9)(10,12)(13,16,18,29)(14,20,22,27)(15,24,26,28)(17,30,31,42)(19,33,35,36)(21,37,38,41)(23,39,25,40)(32,43,34,44);(5,7,8,11)(6,10,9,12)(15,23)(17,31)(19,32)(21,38)(24,39)(25,26)(28,40)(30,42)(33,43)(34,35)(36,44)(37,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6592323);