#4440146

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

1770 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)(5,6)(7,12)(8,10)(9,11)(15,19)(17,20)(21,22,26,27)(23,25,29,35)(24,31,33,28)(30,32,36,34);(5,7,9,8)(6,10,11,12)(13,14,16,18)(15,17,19,20)(22,27)(23,29)(28,31)(30,36);(1,2,4,3)(21,23)(22,25)(24,30)(26,29)(27,35)(28,34)(31,32)(33,36);(1,3,4,2)(5,6)(7,10)(8,12)(9,11)(13,15,16,19)(14,17,18,20)(21,24,26,33)(22,28,27,31)(23,30,29,36)(25,34,35,32);(5,8,9,7)(6,10,11,12)(21,25)(22,23)(24,32)(26,35)(27,29)(28,36)(30,31)(33,34)

Polycyclic presentation:

2^9|a^2=h,b^2=fgi,c^2=g,d^2=i,e^2=fh,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, 4347342);