#6385015

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1859 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ci,d^b=df,e^b=ei,g^c=gi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6292211);