#7561871

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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

2243 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)(15,20)(16,27)(17,30)(18,34)(19,23)(21,33)(22,28)(24,37)(25,39)(26,36)(29,42)(31,35)(32,43)(38,40)(41,44);(1,2)(3,4)(6,10)(11,12)(13,15,19,26)(14,20,23,36)(16,28,25,18)(17,31,32,29)(21,37,38,44)(22,39,34,27)(24,40,41,33)(30,35,43,42);(5,7,9,8)(6,11,10,12)(13,16,18,15,19,25,28,26)(14,20,22,27,23,36,34,39)(17,29,33,24,32,31,40,41)(21,35,30,37,38,42,43,44);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,33,19,32,28,40)(14,21,22,30,23,38,34,43)(15,24,25,31,26,41,16,29)(20,35,27,37,36,42,39,44);(6,10)(11,12)(17,32)(21,38)(24,41)(29,31)(30,43)(33,40)(35,42)(37,44)

Polycyclic presentation:

2^9|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=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7469067);