#474303

order	512
nilpotency class	3
isoclinism	isoclinic class of #455822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

633 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efgh,c^2=i,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,d^c=de,h^c=hi,g^d=gi,h^d=hi

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, 381499);