#6388340

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1719 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)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22)(23,32)(24,29)(25,30)(26,39)(27,31)(28,34)(33,47)(35,42)(36,44)(37,45)(38,46)(40,43)(41,50)(48,52)(49,51);(1,3,4,2)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(24,36)(29,44)(33,48)(37,38)(41,51)(45,46)(47,52)(49,50);(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,23,27,35)(22,28,31,43)(24,33,36,48)(25,34,26,40)(29,41,44,51)(30,42,39,32)(37,49,38,50)(45,47,46,52);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,24)(22,29)(23,33)(25,38)(26,37)(27,36)(28,41)(30,46)(31,44)(32,47)(34,50)(35,48)(39,45)(40,49)(42,52)(43,51);(6,14)(11,15)(12,13)(19,20)(21,25,27,26)(22,30,31,39)(23,34,35,40)(24,37,36,38)(28,42,43,32)(29,45,44,46)(33,49,48,50)(41,47,51,52)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,g^c=gi,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, 6295536);