#7185516

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1792 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092712);