#401029

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1686 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)(5,6)(7,8)(9,10,16,21)(11,22,12,18)(13,27,30,40)(14,31,15,20)(17,26,33,24)(19,29,36,28)(23,35,25,32)(34,38,37,39);(2,4)(9,11,15,26,16,12,14,24)(10,17,31,18,21,33,20,22)(13,25,29,38,30,23,28,39)(19,35,40,37,36,32,27,34);(2,4)(9,12)(10,18)(11,16)(13,25)(14,26)(15,24)(17,20)(19,34)(21,22)(23,30)(27,32)(28,39)(29,38)(31,33)(35,40)(36,37);(6,8)(9,13,14,28,16,30,15,29)(10,19,20,27,21,36,31,40)(11,23,24,38,12,25,26,39)(17,32,22,37,33,35,18,34)

Polycyclic presentation:

2^9|a^2=fi,b^2=efi,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di,g^b=gi,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, 308225);