#7562267

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	#45093	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2214 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=fg,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=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, 7469463);