#6396709

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

1650 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)(23,32)(24,36)(25,30)(26,31)(27,41)(28,35)(29,39)(33,47)(34,44)(37,45)(38,51)(40,43)(42,52)(46,48)(49,50);(1,2,4,3)(5,7)(6,11)(8,15)(9,17)(10,16)(12,19)(13,14)(18,20)(21,23)(22,28)(24,33)(25,40)(26,35)(27,34)(29,42)(30,44)(31,32)(36,50)(37,52)(38,49)(39,48)(41,43)(45,46)(47,51);(1,3,4,2)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24,25,37,26,38,27,39)(22,29,41,51,31,45,30,36)(23,33,40,52,35,49,34,48)(28,42,43,47,32,46,44,50);(7,16)(11,14)(15,17)(19,20)(21,25,26,27)(22,30,31,41)(23,34,35,40)(24,37,38,39)(28,43,32,44)(29,36,45,51)(33,48,49,52)(42,47,46,50);(5,9,10,8)(6,12,13,18)(7,17,16,15)(11,19,14,20)(22,31)(23,35)(24,38)(30,41)(34,40)(37,39)(42,46)(47,50)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fgh,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,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, 6303905);