#494859

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

802 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,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, 402055);