#6508790

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415986);