#415884

order	512
nilpotency class	3
isoclinism	isoclinic class of #394177
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=eh,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 323080);