#1303440

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

2554 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:

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

Polycyclic presentation:

2^9|a^2=fg,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, 1210636);