#497728

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

680 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)(14,20)(15,19)(16,24)(17,30)(21,37)(22,39)(23,36)(25,33)(26,42)(27,29)(28,40)(31,34)(35,41)(38,43);(5,7,8,11)(6,9,10,12)(13,14,18,23)(15,21,19,34)(16,26,29,39)(17,31,33,37)(20,25,36,30)(22,40,42,28)(24,43,27,35)(32,38,44,41);(6,10)(9,12)(13,15,18,19)(14,21,23,34)(16,27,29,24)(17,25,33,30)(20,31,36,37)(22,41,42,38)(26,35,39,43)(28,44,40,32);(13,16,18,29)(14,22,23,42)(15,24,19,27)(17,32,33,44)(20,35,36,43)(21,38,34,41)(25,40,30,28)(26,37,39,31)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,d^c=di,e^d=eh,h^d=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, 404924);