#506513

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×#2186	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

813 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,d^2=i,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, 413709);