#455346

order	512
nilpotency class	3
isoclinism	isoclinic class of #454095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 362542);