#412735

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1094 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319931);