#422832

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

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

1902 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330028);