#6808603

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1684 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)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22)(23,28)(24,29)(25,41)(26,31)(27,32)(30,34)(33,42)(35,43)(36,44)(37,45)(38,46)(39,50)(40,47)(48,51)(49,52);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,35)(22,28,32,43)(24,36,38,39)(25,37,34,26)(29,44,46,50)(30,31,41,45)(33,48,49,40)(42,51,52,47);(2,4)(21,24,37,48,27,38,26,40)(22,29,45,51,32,46,31,47)(23,33,25,39,35,49,34,36)(28,42,41,50,43,52,30,44);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(24,38)(29,46)(33,49)(36,39)(40,48)(42,52)(44,50)(47,51);(7,15)(10,13)(14,20)(18,19)(21,25,26,23,27,34,37,35)(22,30,31,43,32,41,45,28)(24,39,40,33,38,36,48,49)(29,44,47,52,46,50,51,42)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6715799);