#7562271

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

2198 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)(13,14,18,23)(15,21,26,20)(16,28,27,37)(17,31,33,44)(19,35,29,24)(22,39,41,34)(25,42,43,40)(30,36,32,38);(1,2)(3,4)(6,10)(11,12)(13,15,18,26)(14,20,23,21)(16,29,27,19)(17,32,33,30)(22,40,41,42)(24,28,35,37)(25,39,43,34)(31,36,44,38);(5,7,9,8)(6,11,10,12)(13,16,29,26,18,27,19,15)(14,21,35,37,23,20,24,28)(17,32,39,25,33,30,34,43)(22,36,44,42,41,38,31,40);(2,4)(5,8,9,7)(6,12,10,11)(13,17,29,39,18,33,19,34)(14,22,35,44,23,41,24,31)(15,25,16,30,26,43,27,32)(20,36,28,42,21,38,37,40);(6,10)(11,12)(14,23)(17,33)(20,21)(24,35)(25,43)(28,37)(30,32)(34,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,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, 7469467);