#2892825

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
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

1876 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=fhi,d^2=f,e^2=g,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, 2800021);