#495466

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

598 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=fi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,h^b=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, 402662);