#494507

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	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1400 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)(9,10,13,20,29,37,28,36,15,18,12,19,14,21,31,17)(11,22,25,34,32,39,27,16,23,33,24,40,26,38,30,35);(1,2,3,4)(9,11)(10,16)(12,27)(13,30)(14,32)(15,23)(17,33)(18,35)(19,38)(20,39)(21,40)(22,36)(24,28)(25,31)(26,29)(34,37);(6,8)(10,17)(12,28)(13,31)(14,29)(16,33)(18,36)(19,37)(20,21)(22,35)(24,27)(25,30)(26,32)(34,38)(39,40);(10,18)(11,23)(17,36)(19,20)(21,37)(24,25)(26,32)(27,30)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,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, 401703);