#6817549

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgh,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6724745);