#123445

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#380
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,h^a=hi,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30641);