#503518

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

823 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=ef,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410714);