#6388258

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
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

1714 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)(28,35)(33,43)(34,47)(36,44)(37,46)(38,45)(40,41)(42,50)(48,52)(49,51);(1,3,4,2)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(23,33)(28,41)(32,43)(34,48)(35,40)(42,51)(47,52)(49,50);(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,23)(22,28)(24,34)(25,35)(26,40)(27,33)(29,42)(30,43)(31,41)(32,39)(36,49)(37,48)(38,50)(44,47)(45,52)(46,51);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,24,27,37)(22,29,31,46)(23,34,33,48)(25,38,26,36)(28,42,41,51)(30,45,39,44)(32,47,43,52)(35,50,40,49);(6,14)(11,15)(12,13)(19,20)(21,25,27,26)(22,30,31,39)(23,35,33,40)(24,36,37,38)(28,43,41,32)(29,44,46,45)(34,49,48,50)(42,47,51,52)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6295454);