#5610840

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
automorphism	Aut(G) order 1048576
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2890 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)(5,6,9,7)(8,11,12,10)(14,15)(16,20)(21,22,26,24)(23,28,25,27);(5,6,9,7)(8,11,12,10)(13,14,17,15)(16,19,20,18)(21,22,26,24)(23,27,25,28);(6,7)(10,11)(13,14,17,15)(16,18,20,19)(21,23,26,25)(22,27,24,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,18,20,19)(21,24,26,22)(23,27,25,28);(5,8,9,12)(6,10,7,11)(13,16)(14,18)(15,19)(17,20)(21,25,26,23)(22,27,24,28)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=gh,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300351, #72300391, #72300407
maximal normal subgroup of	#72300351, #72300391, #72300407

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5518036);