#473851

order	512
nilpotency class	3
isoclinism	isoclinic class of #460330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

869 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)(3,6)(4,5)(7,8)(9,10,16,21,14,20,15,29)(11,22,24,33,25,17,31,39)(12,18,30,36,23,35,27,38)(13,19,26,40,28,37,32,34);(2,5)(6,8)(9,11,16,31,14,25,15,24)(10,17,21,33,20,22,29,39)(12,26,30,13,23,32,27,28)(18,34,36,37,35,40,38,19);(1,3)(2,6)(4,7)(5,8)(9,12,14,23)(10,18,20,35)(11,13,25,28)(15,30,16,27)(17,19,22,37)(21,38,29,36)(24,32,31,26)(33,40,39,34);(3,7)(6,8)(9,13)(10,19)(11,23)(12,25)(14,28)(15,26)(16,32)(17,18)(20,37)(21,34)(22,35)(24,30)(27,31)(29,40)(33,36)(38,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,g^c=gi,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, 381047);