#413101

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

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

1528 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=e,d^2=hi,e^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,d^b=di,f^c=fi,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, 320297);