#475047

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1109 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,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, 382243);