#493992

order	512
nilpotency class	4
isoclinism	isoclinic class of #480263
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×Q₈	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1053 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^c=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, 401188);