#417852

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2732 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)(9,10,15,20)(11,21,24,40)(12,25,28,36)(13,18,22,34)(14,31,32,19)(16,33,35,23)(17,26,37,27)(29,39,30,38);(1,2)(3,4)(9,11,14,23,15,24,32,33)(10,16,19,21,20,35,31,40)(12,13,26,29,28,22,27,30)(17,18,36,38,37,34,25,39);(2,4)(9,12)(10,17)(11,22)(13,24)(14,26)(15,28)(16,34)(18,35)(19,36)(20,37)(21,39)(23,30)(25,31)(27,32)(29,33)(38,40);(6,8)(9,13,15,22)(10,18,20,34)(11,12,24,28)(14,30,32,29)(16,17,35,37)(19,39,31,38)(21,25,40,36)(23,27,33,26)

Polycyclic presentation:

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

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, 325048);