#7149196

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	#45196	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1889 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)(13,14,19,23,18,20,17,30)(15,24,26,33,27,36,32,21)(16,22,28,38,29,34,31,37)(25,39,42,43,41,44,40,35);(1,2,3,4)(6,9)(7,11)(14,20)(21,33)(22,34)(23,30)(24,36)(35,43)(37,38)(39,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,27)(14,21,20,33)(16,25,29,41)(17,26,19,32)(22,35,34,43)(23,36,30,24)(28,40,31,42)(37,39,38,44);(2,4)(6,9)(10,12)(13,16)(14,22)(15,25)(17,31)(18,29)(19,28)(20,34)(21,35)(23,38)(24,39)(26,42)(27,41)(30,37)(32,40)(33,43)(36,44);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,20,30)(15,26,27,32)(16,28,29,31)(21,36,33,24)(22,37,34,38)(25,40,41,42)(35,39,43,44)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,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, 7056392);