#6389507

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
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×#466	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1852 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)(15,20)(16,28)(17,22)(18,32)(19,23)(21,30)(24,37)(25,34)(26,35)(27,41)(29,36)(31,42)(33,40)(38,43)(39,44);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,24)(17,25)(18,27)(19,26)(21,33)(22,34)(23,35)(28,37)(29,38)(30,40)(31,39)(32,41)(36,43)(42,44);(13,16,17,29,19,31,18,30)(14,21,32,42,23,36,22,28)(15,24,25,38,26,39,27,40)(20,33,41,44,35,43,34,37);(6,9)(10,12)(13,17,19,18)(14,22,23,32)(15,25,26,27)(16,29,31,30)(20,34,35,41)(21,28,36,42)(24,38,39,40)(33,37,43,44);(5,7,8,11)(6,10,9,12)(15,26)(20,35)(24,39)(25,27)(33,43)(34,41)(37,44)(38,40)

Polycyclic presentation:

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

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, 6296703);