#7515774

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#23975	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1844 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)(7,12)(8,10)(9,11)(13,14)(15,19)(16,28)(17,20)(18,23)(21,31)(22,32)(24,35)(25,37)(26,40)(27,36)(29,34)(30,38)(33,39)(41,43)(42,44);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,27)(14,19,23,36)(16,29,30,41)(17,24,32,26)(20,35,22,40)(21,37,39,44)(25,33,42,31)(28,34,38,43);(5,8,9,7)(6,10,11,12)(14,20)(15,24)(16,30)(17,32)(21,38)(22,23)(25,41)(26,27)(28,39)(29,42)(34,43)(35,40);(2,4)(13,16,18,30)(14,21,23,39)(15,25,27,42)(17,33,32,31)(19,34,36,43)(20,28,22,38)(24,29,26,41)(35,44,40,37);(6,11)(10,12)(16,30)(21,39)(25,42)(28,38)(29,41)(31,33)(34,43)(37,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=i,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,d^c=di,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7422970);