#412733

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
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

1104 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)(12,29,32,39)(13,33,30,38)(14,21,15,20)(17,27,37,26)(18,31,35,25)(19,34,40,24);(5,6,8,7)(9,11,14,26,16,28,15,27)(10,17,20,23,22,37,21,36)(12,30,31,34,32,13,25,24)(18,38,39,19,35,33,29,40);(2,4)(5,7,8,6)(9,12,14,31,16,32,15,25)(10,18,20,39,22,35,21,29)(11,24,26,30,28,34,27,13)(17,19,23,33,37,40,36,38);(9,13,15,34,16,30,14,24)(10,19,21,38,22,40,20,33)(11,25,27,32,28,31,26,12)(17,35,36,39,37,18,23,29)

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=e,d^2=hi,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,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319929);