#409034

order	512
nilpotency class	3
isoclinism	isoclinic class of #395151
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1398 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=f,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 316230);