#438975

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
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	#23393	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1482 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)(6,7)(9,10,15,20)(11,16,25,33)(12,17,27,36)(13,19,14,29)(18,28,35,26)(21,30,37,31)(22,34,38,40)(23,32,24,39);(2,4)(5,6)(7,8)(9,11,13,23,15,25,14,24)(10,16,19,32,20,33,29,39)(12,22,26,37,27,38,28,21)(17,34,18,31,36,40,35,30);(1,2,3,4)(9,12)(10,17)(11,21)(13,28)(14,26)(15,27)(16,30)(18,29)(19,35)(20,36)(22,24)(23,38)(25,37)(31,33)(32,40)(34,39);(5,7,8,6)(9,12)(10,18)(11,22)(13,26)(14,28)(15,27)(16,31)(17,29)(19,36)(20,35)(21,24)(23,37)(25,38)(30,33)(32,40)(34,39)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 346171);