#7614487

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	#45269	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1840 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)(13,14,19,24)(15,25,28,36)(16,30,33,37)(17,23,18,22)(20,29,35,26)(21,31,38,32)(27,41,42,34)(39,43,44,40);(2,4)(6,10)(11,12)(15,26)(16,31)(17,18)(20,25)(21,37)(22,23)(28,29)(30,38)(32,33)(35,36)(39,44)(40,43);(5,7,9,8)(6,11,10,12)(13,15,18,29,19,28,17,26)(14,20,23,36,24,35,22,25)(16,27,31,43,33,42,32,40)(21,34,37,44,38,41,30,39);(1,2,3,4)(13,16,18,31,19,33,17,32)(14,21,23,37,24,38,22,30)(15,27,29,43,28,42,26,40)(20,34,36,44,35,41,25,39);(5,8,9,7)(6,11,10,12)(15,28)(20,35)(25,36)(26,29)(27,42)(34,41)(39,44)(40,43)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=h,e^2=f,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, 7521683);