#501329

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

735 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)(7,11)(8,10)(9,12)(13,14,21,36,17,32,35,44,18,25,19,26,20,24,33,38)(15,27,31,40,30,41,37,42,16,22,29,43,28,39,34,23);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,18,16)(14,22,25,27)(17,30,20,28)(19,29,21,31)(23,38,42,44)(24,41,32,39)(26,40,36,43)(33,34,35,37);(2,4)(13,16,18,15)(14,23,25,42)(17,30,20,28)(19,34,21,37)(22,38,27,44)(24,40,32,43)(26,41,36,39)(29,35,31,33);(6,10)(9,11)(13,17,18,20)(14,24,25,32)(15,28,16,30)(19,33,21,35)(22,39,27,41)(23,40,42,43)(26,44,36,38)(29,37,31,34)

Polycyclic presentation:

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

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, 408525);