#4597886

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

3340 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)(6,9)(7,11)(13,14)(15,20)(16,19)(17,18)(22,25)(23,24);(6,9)(10,12)(13,15,18,16)(14,19,17,20)(21,22,26,25)(23,27,24,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,16)(14,20,17,19)(21,23,26,24)(22,27,25,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,20,17,19)(21,24,26,23)(22,27,25,28);(5,7)(6,10)(8,11)(9,12)(13,17,18,14)(15,20,16,19)(21,25,26,22)(23,27,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4505082);