#506505

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2184	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

883 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413701);