#2120917

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2123 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)(5,6)(7,12)(8,9)(10,11)(13,14,17,19)(15,20,18,16)(21,22,26,23)(24,27,31,29)(25,28,33,30)(32,34,36,35);(5,6,10,11)(7,12,8,9)(13,15,17,18)(14,16,19,20)(22,23)(24,31)(28,30)(32,36);(5,7,10,8)(6,9,11,12)(13,14)(15,16)(17,19)(18,20)(21,23,26,22)(24,29,31,27)(25,28,33,30)(32,34,36,35);(1,2)(3,4)(5,8,10,7)(6,9,11,12)(21,24)(22,27)(23,29)(25,32)(26,31)(28,34)(30,35)(33,36);(5,9)(6,7)(8,11)(10,12)(13,16)(14,18)(15,19)(17,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=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, 2028113);