#7142945

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#45332	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1991 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)(7,11)(10,12)(13,14,18,22)(15,23,26,36)(16,27,28,20)(17,30,32,21)(19,31,34,25)(24,38,39,37)(29,35,43,42)(33,40,41,44);(1,2)(3,4)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,22,34)(16,24,28,39)(17,31,32,25)(20,33,27,41)(21,36,30,23)(29,44,43,40)(35,38,42,37);(2,4)(6,9)(10,12)(13,16,18,28)(14,20,22,27)(15,24,26,39)(17,29,32,43)(19,33,34,41)(21,35,30,42)(23,37,36,38)(25,40,31,44);(5,7,8,11)(6,10,9,12)(16,28)(20,27)(24,39)(29,43)(33,41)(35,42)(37,38)(40,44)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050141);