#497849

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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×D₄	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

664 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fhi,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405045);