#504764

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

586 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)(13,14,19,33,31,40,29,20,18,24,16,22,17,23,34,36)(15,25,42,37,32,44,30,21,28,43,26,41,27,39,35,38);(1,2)(3,4)(6,9)(10,11)(14,20)(16,29)(17,31)(19,34)(21,25)(22,23)(24,36)(26,30)(27,32)(33,40)(35,42)(37,44)(38,43)(39,41);(2,4)(13,15,18,28)(14,21,24,38)(16,30,19,35)(17,32,31,27)(20,25,36,43)(22,39,33,44)(23,41,40,37)(26,34,42,29);(5,7,8,12)(6,10,9,11)(14,20)(16,29)(17,31)(19,34)(21,25)(22,23)(24,36)(26,30)(27,32)(33,40)(35,42)(37,44)(38,43)(39,41)

Polycyclic presentation:

2^9|a^2=ei,c^2=i,d^2=efh,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, 411960);