#7062290

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	#42995	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1867 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)(5,6)(7,9)(8,10)(11,12)(13,14,18,22,19,23,17,31)(15,24,27,34,25,36,33,20)(16,21,30,40,28,37,32,39)(26,41,43,38,29,44,42,35);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(15,25)(16,28)(20,34)(21,37)(24,36)(27,33)(30,32)(39,40);(5,7,8,12)(6,9,10,11)(13,15,19,25)(14,20,23,34)(16,29,28,26)(17,27,18,33)(21,38,37,35)(22,36,31,24)(30,43,32,42)(39,44,40,41);(2,4)(7,12)(9,11)(13,16,19,28)(14,21,23,37)(15,26,25,29)(17,32,18,30)(20,35,34,38)(22,40,31,39)(24,41,36,44)(27,43,33,42);(6,10)(9,11)(13,17,19,18)(14,22,23,31)(15,27,25,33)(16,30,28,32)(20,36,34,24)(21,39,37,40)(26,42,29,43)(35,41,38,44)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,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, 6969486);