#6673873

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1720 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)(21,22,25,27)(23,31,32,44)(24,35,36,29)(26,39,38,30)(28,40,41,34)(33,47,46,45)(37,43,51,50)(42,48,49,52);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(22,27)(23,32)(24,36)(30,39)(34,40)(37,51)(42,49)(45,47);(2,4)(21,23,25,32)(22,28,27,41)(24,33,36,46)(26,40,38,34)(29,42,35,49)(30,44,39,31)(37,52,51,48)(43,47,50,45);(6,11)(10,13)(12,19)(18,20)(21,24)(22,29)(23,33)(25,36)(26,37)(27,35)(28,42)(30,43)(31,45)(32,46)(34,48)(38,51)(39,50)(40,52)(41,49)(44,47);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(24,36)(29,35)(33,46)(37,51)(42,49)(43,50)(45,47)(48,52)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6581069);