#7521356

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1780 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)(15,21)(16,29)(17,33)(18,24)(19,35)(20,25)(22,32)(23,30)(26,42)(27,39)(28,41)(31,38)(34,43)(36,40)(37,44);(1,2)(3,4)(6,10)(11,12)(13,15,20,28)(14,21,25,41)(16,30,26,37)(17,31,34,32)(18,27,19,36)(22,33,38,43)(23,42,44,29)(24,39,35,40);(13,16,20,26)(14,22,25,38)(15,17,28,34)(18,32,19,31)(21,23,41,44)(24,42,35,29)(27,37,36,30)(33,40,43,39);(2,4)(5,7,9,8)(6,11,10,12)(13,17)(14,23)(15,26)(16,28)(18,30)(19,37)(20,34)(21,38)(22,41)(24,43)(25,44)(27,32)(29,40)(31,36)(33,35)(39,42);(5,8,9,7)(6,11,10,12)(13,18,20,19)(14,24,25,35)(15,27,28,36)(16,31,26,32)(17,30,34,37)(21,39,41,40)(22,29,38,42)(23,43,44,33)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7428552);