#494860

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

882 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,h^a=hi,d^b=di,f^c=fh,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, 402056);