#6687779

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1696 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=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, 6594975);