#6384456

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	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2079 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)(6,9)(10,12)(13,14,19,23)(15,20,26,36)(16,28,31,38)(17,22,18,32)(21,33,39,30)(24,40,29,44)(25,35,27,42)(34,43,37,41);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,29)(17,25)(18,27)(19,26)(21,37)(22,35)(23,36)(24,31)(28,44)(30,43)(32,42)(33,41)(34,39)(38,40);(5,7,8,11)(6,10,9,12)(13,16,19,31)(14,21,23,39)(15,24,26,29)(17,30,18,33)(20,34,36,37)(22,38,32,28)(25,41,27,43)(35,44,42,40);(6,9)(10,12)(13,17,19,18)(14,22,23,32)(15,25,26,27)(16,30,31,33)(20,35,36,42)(21,38,39,28)(24,41,29,43)(34,44,37,40);(1,3,4,2)(15,26)(16,31)(20,36)(21,39)(25,27)(28,38)(30,33)(35,42)

Polycyclic presentation:

2^9|a^2=h,c^2=fi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^c=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, 6291652);