#462314

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
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

863 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)(11,19)(12,22)(13,25)(15,27)(17,33)(20,36)(21,26)(23,35)(28,38)(29,34)(30,39)(32,40);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,27)(20,28)(21,29)(22,31)(23,30)(24,32)(25,33)(26,34)(35,38)(36,39)(37,40);(3,7)(5,8)(9,11,14,21)(10,15,18,29)(12,23,22,20)(13,26,25,19)(16,30,31,28)(17,34,33,27)(24,35,37,36)(32,38,40,39);(9,12,13,24,14,22,25,37)(10,16,17,32,18,31,33,40)(11,20,19,35,21,23,26,36)(15,28,27,38,29,30,34,39)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=f,f^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,h^d=hi

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, 369510);