#509573

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

512 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=e,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, 416769);