#509076

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

712 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416272);