#6815870

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	#42673	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2153 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

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, 6723066);