#7104376

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1681 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)(3,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,28,42,27,25,26,31)(23,32,34,29,36,44,35,41)(24,30,38,45,40,37,39,46)(33,47,50,43,49,52,48,51);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,36)(22,29,25,41)(24,33,40,49)(26,34,28,35)(30,43,37,51)(31,44,42,32)(38,48,39,50)(45,47,46,52);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24)(22,30)(23,33)(25,37)(26,39)(27,40)(28,38)(29,43)(31,46)(32,47)(34,50)(35,48)(36,49)(41,51)(42,45)(44,52);(8,16)(11,20)(13,15)(17,19)(21,25,28,31,27,22,26,42)(23,32,34,29,36,44,35,41)(24,37,38,46,40,30,39,45)(33,47,50,43,49,52,48,51);(6,12)(7,14)(11,20)(13,15)(21,26,27,28)(22,31,25,42)(23,34,36,35)(24,38,40,39)(29,44,41,32)(30,45,37,46)(33,48,49,50)(43,47,51,52)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=h,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7011572);