#493971

order	512
nilpotency class	4
isoclinism	isoclinic class of #480274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1079 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=di,f^c=fi,g^c=ghi,g^f=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, 401167);