#6395348

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1758 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)(16,29)(17,22)(18,23)(19,32)(20,27)(21,30)(25,40)(26,39)(28,35)(31,36)(33,37)(34,43)(38,41)(42,44);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,27)(14,20,23,24)(16,25,31,42)(17,26,19,28)(21,34,37,38)(22,35,32,39)(29,44,36,40)(30,41,33,43);(1,3,4,2)(13,16)(14,21)(15,25)(17,30)(18,31)(19,33)(20,34)(22,36)(23,37)(24,38)(26,41)(27,42)(28,43)(29,32)(35,40)(39,44);(6,9)(10,12)(13,17,18,19)(14,22,23,32)(15,26,27,28)(16,30,31,33)(20,35,24,39)(21,36,37,29)(25,41,42,43)(34,40,38,44);(5,7,8,11)(6,10,9,12)(14,23)(15,27)(21,37)(22,32)(25,42)(26,28)(29,36)(41,43)

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,e^b=ei,g^c=gi

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, 6302544);