#4437172

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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

1720 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=g,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,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, 4344368);