#494502

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1424 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^b=di,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401698);