#7006669

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	#42806	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2012 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6913865);