#6395435

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1737 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)(7,11)(13,14)(15,24)(16,21)(17,22)(18,20)(19,26)(23,30)(25,37)(27,33)(28,35)(29,34)(31,40)(32,38)(36,41)(39,43)(42,44);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,17,26)(14,19,22,24)(16,25,28,38)(18,27,30,40)(20,31,23,33)(21,32,35,37)(29,39,41,44)(34,42,36,43);(1,3,4,2)(14,20)(18,30)(19,31)(21,34)(22,23)(24,33)(27,40)(29,41)(32,42)(35,36)(37,43)(39,44);(6,9)(10,12)(13,16,17,28)(14,21,22,35)(15,25,26,38)(18,29,30,41)(19,32,24,37)(20,34,23,36)(27,39,40,44)(31,42,33,43);(5,7,8,11)(6,10,9,12)(14,22)(16,28)(19,24)(20,23)(25,38)(29,41)(31,33)(39,44)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6302631);