#6675185

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1789 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)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22)(23,31)(24,35)(25,29)(26,39)(27,33)(28,37)(30,38)(32,47)(34,42)(36,44)(40,43)(41,51)(45,49)(46,50)(48,52);(1,2)(3,4)(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(21,23)(22,27)(24,32)(25,33)(26,34)(28,41)(29,31)(30,42)(35,52)(36,48)(37,49)(38,50)(39,46)(40,51)(43,45)(44,47);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,24)(22,28)(23,32)(25,36)(26,40)(27,41)(29,43)(30,44)(31,45)(33,48)(34,51)(35,39)(37,38)(42,47)(46,52)(49,50);(6,13)(11,15)(12,14)(19,20)(22,29)(23,33)(28,43)(30,39)(32,48)(34,50)(35,44)(49,51);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(23,33)(24,36)(27,31)(28,43)(34,50)(35,44)(37,40)(42,46)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,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,e^c=ei,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582381);