#6817022

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2109 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724218);