#7062234

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

1845 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)(13,14,18,23,19,24,15,20)(16,27,25,35,29,38,32,21)(17,22,30,36,31,37,26,33)(28,40,39,44,42,43,41,34);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,19,18)(14,20,24,23)(16,25,29,32)(17,30,31,26)(21,27,35,38)(22,36,37,33)(28,41,42,39)(34,43,44,40);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,25)(17,28)(18,32)(19,29)(20,27)(22,34)(23,38)(24,35)(26,39)(30,41)(31,42)(33,40)(36,43)(37,44);(2,4)(7,12)(9,11)(13,17,19,31)(14,22,24,37)(15,26,18,30)(16,28,29,42)(20,33,23,36)(21,34,35,44)(25,39,32,41)(27,40,38,43);(6,10)(9,11)(13,15,19,18)(14,20,24,23)(16,25,29,32)(17,30,31,26)(21,27,35,38)(22,36,37,33)(28,41,42,39)(34,43,44,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,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=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, 6969430);