#497779

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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 ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,22,31,39,33,17,19,23,36,41,16,21,30,34)(15,24,27,37,25,42,43,26,29,32,44,38,28,40,35,20);(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,25)(17,32)(18,35)(19,29)(21,37)(22,40)(23,26)(24,34)(27,30)(28,31)(33,44)(36,43)(38,39)(41,42);(6,10)(9,12)(13,16,19,31)(14,21,23,39)(15,25,29,28)(17,22,34,41)(18,30,36,33)(20,37,26,38)(24,42,32,40)(27,43,44,35);(13,17,33,21,16,22,18,23,19,34,30,39,31,41,36,14)(15,26,43,40,28,37,27,32,29,20,35,42,25,38,44,24)

Polycyclic presentation:

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

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, 404975);