#5026218

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 2097152
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2182 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4933414);