#6517128

order	512
nilpotency class	3
isoclinism	isoclinic class of #6502828
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(9,10,14,18,15,19,13,29)(11,20,24,32,22,37,23,16)(12,25,28,34,27,35,30,17)(21,31,38,40,26,33,39,36);(1,2)(3,6)(4,5)(7,8)(9,11,13,23,15,22,14,24)(10,16,18,20,19,32,29,37)(12,26,28,39,27,21,30,38)(17,33,35,40,34,31,25,36);(2,5)(3,7)(9,12)(10,17)(11,21)(13,28)(14,30)(15,27)(16,31)(18,35)(19,34)(20,36)(22,26)(23,38)(24,39)(25,29)(32,33)(37,40);(1,3,4,7)(2,6,5,8)(11,22)(12,27)(16,32)(17,34)(20,37)(23,24)(25,35)(28,30);(2,5)(6,8)(11,22)(16,32)(20,37)(21,26)(23,24)(31,33)(36,40)(38,39)

Polycyclic presentation:

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

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, 6424324);