#469340

order	512
nilpotency class	3
isoclinism	isoclinic class of #454982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1250 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,25)(13,21)(14,19)(15,20)(17,26)(18,23)(22,34)(24,32)(27,35)(28,36)(29,31)(30,37)(33,39)(38,40);(2,5)(6,8)(9,11)(10,16)(12,22)(13,29)(14,23)(15,24)(17,33)(18,19)(20,32)(21,31)(25,34)(26,39)(27,37)(28,38)(30,35)(36,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,28)(10,17,20,35)(11,22,24,38)(13,27,14,26)(16,30,32,33)(18,34,31,40)(19,36,21,25)(23,39,29,37);(3,7)(6,8)(9,11)(10,18,20,31)(12,22)(13,29)(14,23)(15,24)(16,19,32,21)(17,34,35,40)(25,30,36,33)(26,39)(27,37)(28,38)

Polycyclic presentation:

2^9|c^2=i,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,f^c=fi,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, 376536);