#4308438

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 16777216
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

2226 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)(7,11)(10,12)(13,14,17,15)(16,19,20,18)(23,28)(24,26);(6,9)(10,12)(14,15)(18,19)(21,22,25,27)(23,26,28,24);(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,27)(25,28);(1,2,3,4)(21,24,25,26)(22,23,27,28);(5,7,8,11)(6,10,9,12)(13,16,17,20)(14,18,15,19)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

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, 4215634);