#403818

order	512
nilpotency class	3
isoclinism	isoclinic class of #393934
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egi,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^c=hi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311014);