#465861

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1032 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,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, 373057);