#1303341

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

2048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(5,6,8,10)(7,11,12,9)(13,14,18,20)(15,16,19,17)(21,22)(23,28)(24,32)(25,30)(26,27)(29,34)(31,35)(33,36);(1,2,3,4)(13,15,18,19)(14,16,20,17)(22,27)(23,30)(29,32)(31,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,23,26,30)(22,28,27,25)(24,31,34,36)(29,33,32,35);(13,16,18,17)(14,15,20,19)(21,24,26,34)(22,29,27,32)(23,31,30,36)(25,35,28,33);(5,7,8,12)(6,9,10,11)(13,17,18,16)(14,19,20,15)(21,25,26,28)(22,30,27,23)(24,33,34,35)(29,31,32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=f,d^2=i,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, 1210537);