#497489

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, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#547 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

978 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ei,d^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,d^c=di,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, 404685);