#6671832

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37682	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6579028);