#503520

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	#33179	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

852 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)(14,20)(15,24)(16,29)(17,31)(18,25)(21,35)(22,30)(23,34)(26,28)(27,41)(32,36)(37,44)(38,39)(40,43);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,14,17,20,19,23,31,34)(15,21,27,32,28,37,39,43)(16,30,29,42,18,22,25,33)(24,36,38,35,26,40,41,44);(2,4)(5,7,9,8)(6,11,12,10)(13,15,25,38,17,27,16,26,19,28,29,41,31,39,18,24)(14,21,33,35,20,32,30,40,23,37,42,44,34,43,22,36);(5,8,9,7)(6,11,12,10)(13,16,31,25,19,18,17,29)(14,22,34,42,23,30,20,33)(15,26,39,38,28,24,27,41)(21,36,43,44,37,40,32,35)

Polycyclic presentation:

2^9|a^2=ehi,b^2=e,c^2=ef,d^2=ei,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, 410716);