#509968

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

525 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=ei,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,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, 417164);