#416246

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1260 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323442);