#494201

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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

584 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)(15,23)(16,19)(17,30)(18,31)(20,26)(21,25)(22,29)(24,35)(27,42)(28,43)(32,39)(33,38)(34,44)(36,41)(37,40);(6,9)(10,12)(14,19)(15,24)(21,33)(22,31)(26,39)(27,40)(30,43)(41,44);(13,15,25,41,18,27,28,39,16,24,38,44,29,40,17,26)(14,20,30,37,22,34,33,35,19,32,43,42,31,36,21,23);(5,7,8,11)(6,10,9,12)(15,24)(20,32)(23,35)(26,39)(27,40)(34,36)(37,42)(41,44)

Polycyclic presentation:

2^9|a^2=g,c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,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, 401397);