#6384444

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2034 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)(13,14,18,23)(15,19,24,32)(16,21,27,35)(17,20,29,22)(25,31,37,42)(26,33,38,30)(28,34,40,36)(39,43,44,41);(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,19,23,32)(16,25,27,37)(17,26,29,38)(20,33,22,30)(21,31,35,42)(28,39,40,44)(34,43,36,41);(5,7,8,11)(6,10,9,12)(14,20)(15,24)(17,29)(19,30)(21,34)(22,23)(25,37)(28,40)(31,41)(32,33)(35,36)(42,43);(6,9)(10,12)(13,16,18,27)(14,21,23,35)(15,25,24,37)(17,28,29,40)(19,31,32,42)(20,34,22,36)(26,39,38,44)(30,41,33,43);(1,3,4,2)(15,24)(16,27)(19,32)(21,35)(26,38)(28,40)(30,33)(34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,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,g^c=gi,e^d=ei

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, 6291640);