#509874

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

503 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)(7,8)(9,11)(10,12)(13,14,20,35,15,21,32,22,18,25,17,24,19,26,28,38)(16,29,42,37,34,44,33,23,31,43,30,41,27,40,36,39);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,19)(14,21,25,26)(16,27,31,34)(17,28,20,32)(22,24,38,35)(23,37,39,41)(29,40,43,44)(30,33,42,36);(2,4)(5,8,9,10)(6,7,11,12)(14,22)(15,19)(17,32)(20,28)(21,35)(23,29)(24,26)(25,38)(27,34)(30,33)(36,42)(37,44)(39,43)(40,41);(6,11)(7,12)(13,16,18,31)(14,23,25,39)(15,27,19,34)(17,33,20,36)(21,37,26,41)(22,29,38,43)(24,40,35,44)(28,42,32,30)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,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, 417070);