#7614498

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1822 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=hi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521694);