#509310

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

629 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)(7,8)(9,11)(10,12)(13,14,16,22)(15,25,26,19)(17,20,29,21)(18,32,30,24)(23,38,36,31)(27,37,41,33)(28,35,42,40)(34,43,39,44);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,26)(17,27)(18,28)(20,33)(21,37)(22,25)(23,34)(24,35)(29,41)(30,42)(31,43)(32,40)(36,39)(38,44);(2,4)(5,8,9,10)(6,7,11,12)(14,20)(17,31)(18,30)(19,33)(21,22)(23,24)(25,37)(27,43)(28,42)(29,38)(32,36)(34,35)(39,40)(41,44);(6,11)(7,12)(14,21)(15,26)(17,31)(18,30)(19,33)(20,22)(23,32)(24,36)(25,37)(27,44)(29,38)(34,35)(39,40)(41,43)

Polycyclic presentation:

2^9|a^2=i,c^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=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, 416506);