#4414424

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=gh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef

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, 4321620);