#504765

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

594 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:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=i,d^2=efhi,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi,e^d=eh,f^d=fh,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, 411961);