#396168

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,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, 303364);