#2580680

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1949 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)(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,22)(23,28)(24,31)(25,27)(26,33)(29,34)(30,36)(32,35);(1,2)(3,4)(13,15)(14,16)(17,20)(18,19)(21,23,27,24)(22,28,25,31)(26,30,34,32)(29,35,33,36);(2,4)(14,18)(16,19)(21,24,27,23)(22,28,25,31)(26,32,34,30)(29,35,33,36);(5,7,9,12)(6,10,8,11)(13,15)(14,16)(17,20)(18,19)(21,25,27,22)(23,28,24,31)(26,33,34,29)(30,35,32,36);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,16)(14,15)(17,19)(18,20)(21,26)(22,29)(23,30)(24,32)(25,33)(27,34)(28,35)(31,36)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=fg,d^2=fhi,e^2=ghi,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, 2487876);