#6389693

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1685 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(1,2,4,3)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22,25,27)(23,31,33,44)(24,29,35,40)(26,38,37,30)(28,41,39,34)(32,45,46,52)(36,49,48,43)(42,51,50,47);(1,3,4,2)(5,7)(6,11)(8,15)(9,17)(10,16)(12,19)(13,14)(18,20)(22,27)(24,35)(28,39)(30,38)(31,44)(32,46)(36,48)(47,51);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,23,26,34,25,33,37,41)(22,28,30,31,27,39,38,44)(24,32,36,47,35,46,48,51)(29,42,43,45,40,50,49,52);(7,16)(11,14)(15,17)(19,20)(21,24)(22,29)(23,32)(25,35)(26,36)(27,40)(28,42)(30,43)(31,45)(33,46)(34,47)(37,48)(38,49)(39,50)(41,51)(44,52);(5,9,10,8)(6,12,13,18)(7,17,16,15)(11,19,14,20)(22,27)(23,33)(24,35)(30,38)(34,41)(36,48)(42,50)(45,52)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=fgi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^c=ei,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, 6296889);