#6811797

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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

1656 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)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22)(23,28)(24,29)(25,39)(26,31)(27,32)(30,40)(33,42)(34,43)(35,44)(36,45)(37,48)(38,46)(41,47)(49,51)(50,52);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23,27,34)(22,28,32,43)(24,35,36,37)(25,26,40,41)(29,44,45,48)(30,47,39,31)(33,49,50,38)(42,51,52,46);(2,4)(21,24,26,38,27,36,41,49)(22,29,31,46,32,45,47,51)(23,33,25,35,34,50,40,37)(28,42,39,44,43,52,30,48);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(24,36)(29,45)(33,50)(35,37)(38,49)(42,52)(44,48)(46,51);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,25,26,34,27,40,41,23)(22,30,31,28,32,39,47,43)(24,37,38,33,36,35,49,50)(29,44,46,52,45,48,51,42)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=h,d^2=f,e^2=fh,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,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, 6718993);