#6384640

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
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

1771 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)(6,9)(10,11)(13,14,19,24)(15,20,27,37)(16,28,17,22)(18,32,33,23)(21,31,39,30)(25,35,29,44)(26,41,43,36)(34,42,38,40);(5,6,8,9)(7,11,12,10)(13,15)(14,20)(16,29)(17,25)(18,26)(19,27)(21,38)(22,35)(23,36)(24,37)(28,44)(30,42)(31,40)(32,41)(33,43)(34,39);(13,16,19,17)(14,21,24,39)(15,25,27,29)(18,31,33,30)(20,34,37,38)(22,32,28,23)(26,42,43,40)(35,36,44,41);(5,7,8,12)(6,10,9,11)(13,17,19,16)(14,22,24,28)(15,25,27,29)(18,31,33,30)(20,35,37,44)(21,32,39,23)(26,40,43,42)(34,36,38,41);(1,3,4,2)(16,17)(21,39)(22,28)(25,29)(30,31)(34,38)(35,44)(40,42)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291836);