#2816577

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=g,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=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, 2723773);