#497641

order	512
nilpotency class	4
isoclinism	isoclinic class of #480866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂²×D₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#554 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=ei,d^2=e,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,g^c=gh,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404837);