#414184

order	512
nilpotency class	3
isoclinism	isoclinic class of #395151
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1244 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=e,d^2=ghi,e^2=i,f^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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321380);