#6508749

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1844 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)(23,27)(24,34)(25,38)(26,29)(28,37)(30,40)(31,47)(32,42)(33,50)(35,52)(36,46)(39,45)(41,48)(43,49)(44,51);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,27)(24,35)(25,33)(26,32)(28,44)(29,42)(30,43)(31,36)(34,52)(37,51)(38,50)(39,48)(40,49)(41,45)(46,47);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,26,36)(22,28,29,45)(23,31,32,35)(25,39,40,37)(27,41,42,44)(30,46,38,34)(33,51,49,48)(43,52,50,47);(8,16)(11,20)(13,15)(17,19)(22,29)(23,32)(24,36)(30,38)(33,49)(37,39)(41,44)(47,52);(7,14)(10,18)(13,15)(17,19)(23,32)(27,42)(31,35)(33,49)(41,44)(43,50)(47,52)(48,51)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415945);