#416344

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1306 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)(9,10,13,14)(11,15,12,16)(17,18)(19,25)(20,24)(21,28)(22,27)(23,26)(29,31)(30,32);(5,6)(7,8)(10,14)(15,16)(17,19,22,29,23,30,21,20)(18,24,28,32,26,31,27,25);(2,4)(6,8)(9,11,13,12)(10,15,14,16)(17,20,21,30,23,29,22,19)(18,25,27,31,26,32,28,24);(9,12,13,11)(10,15,14,16)(18,26)(24,31)(25,32)(27,28)

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=e,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 323540);