#468413

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

667 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:

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

Polycyclic presentation:

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

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, 375609);