#467495

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

807 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,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, 374691);