#504359

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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

559 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)(7,10)(8,9)(11,12)(13,14,18,24)(15,21,16,28)(17,22,34,36)(19,35,33,25)(20,29,37,26)(23,42,41,32)(27,43,30,40)(31,39,44,38);(1,2)(3,4)(13,15,18,16)(14,20,24,37)(17,31,34,44)(19,30,33,27)(21,36,28,22)(23,40,41,43)(25,39,35,38)(26,42,29,32);(2,4)(6,9)(10,12)(13,16,18,15)(14,21,24,28)(17,29,34,26)(19,30,33,27)(20,36,37,22)(23,39,41,38)(25,40,35,43)(31,32,44,42);(5,7,8,11)(6,10,9,12)(14,22)(17,32)(19,33)(20,21)(23,25)(24,36)(26,31)(27,30)(28,37)(29,44)(34,42)(35,41)(38,43)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411555);