#476148

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23390	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1339 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:

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

Polycyclic presentation:

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

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, 383344);