#6685961

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1703 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,27,31)(23,32,36,47)(24,29,38,45)(25,30,26,40)(28,34,42,35)(33,46,49,51)(37,43,39,44)(41,50,52,48);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18);(2,4)(21,23,25,34,27,36,26,35)(22,28,40,47,31,42,30,32)(24,33,39,50,38,49,37,48)(29,41,43,51,45,52,44,46);(6,11)(10,13)(12,19)(18,20)(21,24)(22,29)(23,33)(25,39)(26,37)(27,38)(28,41)(30,44)(31,45)(32,46)(34,50)(35,48)(36,49)(40,43)(42,52)(47,51);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,37,38,39)(28,32,42,47)(29,43,45,44)(33,48,49,50)(41,51,52,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6593157);