#6811804

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

1649 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)(23,34)(24,39)(25,43)(26,45)(27,33)(28,31)(29,38)(30,42)(32,40)(35,47)(36,48)(37,49)(41,51)(44,52)(46,50);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23)(22,29)(24,40)(25,36)(26,42)(27,38)(28,46)(30,32)(31,48)(33,34)(35,41)(37,44)(39,45)(43,50)(47,49)(51,52);(2,4)(21,24,28,41,27,42,25,37)(22,30,31,49,33,39,43,51)(23,35,36,40,38,44,46,26)(29,47,50,32,34,52,48,45);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,27,28)(22,31,33,43)(23,36,38,46)(24,41,42,37)(26,44,40,35)(29,48,34,50)(30,51,39,49)(32,52,45,47);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,26,28,35,27,40,25,44)(22,32,31,52,33,45,43,47)(23,37,36,24,38,41,46,42)(29,49,50,39,34,51,48,30)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=h,d^2=f,e^2=fhi,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, 6719000);