#6396518

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
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

1864 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)(16,28)(17,22)(18,23)(19,33)(20,27)(21,29)(25,40)(26,36)(30,37)(31,35)(32,38)(34,41)(39,43)(42,44);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,31)(18,27)(19,26)(21,34)(22,36)(23,24)(28,44)(29,43)(30,42)(32,41)(33,35)(37,40)(38,39);(1,3,4,2)(13,16,19,29,18,30,17,32)(14,21,22,28,23,38,33,37)(15,25,26,43,27,42,31,41)(20,34,36,44,24,39,35,40);(6,9)(10,12)(13,17,18,19)(14,22,23,33)(15,26,27,31)(16,29,30,32)(20,35,24,36)(21,37,38,28)(25,41,42,43)(34,44,39,40);(5,7,8,11)(6,10,9,12)(14,23)(15,27)(16,30)(22,33)(26,31)(29,32)(34,39)(40,44)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,d^c=di,e^c=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, 6303714);