#586710

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fh,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 493906);