#509292

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28247	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

545 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416488);