#1303439

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	#16500	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2670 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=fg,d^2=gi,e^2=fg,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, 1210635);