#504868

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

626 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)(13,14,18,22)(15,23,28,40)(16,29,31,34)(17,32,30,21)(19,33,36,26)(20,24,37,25)(27,44,43,39)(35,42,38,41);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,24,41,17,27,16,26,18,28,25,42,30,43,31,33)(14,19,34,44,21,35,20,23,22,36,29,39,32,38,37,40);(2,4)(6,10)(9,11);(6,10)(9,11)(13,15,25,42,17,27,31,33,18,28,24,41,30,43,16,26)(14,19,29,39,21,35,37,40,22,36,34,44,32,38,20,23)

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,d^2=efh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di

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, 412064);