#1302851

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
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

2742 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=g,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1210047);