#509092

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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

706 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)(13,14,17,20)(15,24,16,27)(18,32,30,43)(19,36,31,23)(21,28,37,25)(22,40,38,35)(26,44,29,41)(33,42,34,39);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(14,20)(21,37)(22,38)(23,36)(24,27)(32,43)(39,42)(41,44);(2,4)(13,15,17,16)(14,21,20,37)(18,33,30,34)(19,29,31,26)(22,41,38,44)(23,42,36,39)(24,43,27,32)(25,40,28,35);(6,10)(7,12)(13,16,17,15)(14,21,20,37)(18,34,30,33)(19,26,31,29)(22,41,38,44)(23,42,36,39)(24,43,27,32)(25,35,28,40)

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=dei,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416288);