#494088

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

698 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)(7,10)(8,9)(11,12)(13,14)(15,27)(16,21)(17,35)(18,38)(19,39)(20,26)(22,29)(23,28)(24,37)(25,36)(30,42)(31,41)(32,40)(33,44)(34,43);(6,9)(10,12)(13,15,28,34,19,31,37,32,20,29,18,30,36,33,17,16)(14,21,35,44,25,42,38,22,26,40,24,41,39,43,23,27);(5,7,8,11)(6,10,9,12)(13,16,17,33,36,30,18,29,20,32,37,31,19,34,28,15)(14,22,23,42,39,44,24,21,26,27,38,43,25,41,35,40);(15,29)(16,32)(21,40)(22,27)(30,34)(31,33)(41,44)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401284);