#4416883

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 33554432
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄³

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

2362 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)(7,10)(8,9)(11,12)(15,19)(17,20)(21,22,24,26)(23,27,28,25);(6,9)(10,11)(13,14,16,18)(15,17,19,20);(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,17)(16,18)(19,20);(21,23,24,28)(22,25,26,27);(5,7,8,12)(6,10,9,11)(13,15)(14,17)(16,19)(18,20)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=g,d^2=i,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, 4324079);