#6395337

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	C₂×#121	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1778 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)(15,25)(16,29)(17,24)(18,23)(19,22)(20,28)(21,30)(26,40)(27,35)(31,38)(32,37)(33,36)(34,41)(39,43)(42,44);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,28)(14,20,23,25)(16,26,31,42)(17,32,19,27)(21,34,36,39)(22,37,24,35)(29,44,38,40)(30,43,33,41);(1,3,4,2)(13,16)(14,21)(15,26)(17,30)(18,31)(19,33)(20,34)(22,29)(23,36)(24,38)(25,39)(27,41)(28,42)(32,43)(35,40)(37,44);(6,9)(10,12)(13,17,18,19)(14,22,23,24)(15,27,28,32)(16,30,31,33)(20,35,25,37)(21,29,36,38)(26,41,42,43)(34,40,39,44);(5,7,8,11)(6,10,9,12)(14,23)(15,28)(21,36)(22,24)(26,42)(27,32)(29,38)(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,d^a=di,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi

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, 6302533);