#4928797

order	512
nilpotency class	2
isoclinism	isoclinic class of #4928752
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2438 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4835993);