#3052863

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

1836 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)(7,8)(9,11)(10,12)(13,14,17,20)(15,18,16,19)(21,22)(23,24)(25,33)(26,27)(28,31)(29,34)(30,35)(32,36);(2,4)(5,7,9,12)(6,10,11,8)(13,15,17,16)(14,18,20,19)(22,27)(24,28)(25,34)(30,36);(1,2)(3,4)(13,15,17,16)(14,18,20,19)(21,23)(22,28)(24,27)(25,30)(26,31)(29,35)(32,33)(34,36);(5,6)(7,10)(8,12)(9,11)(13,14)(15,18)(16,19)(17,20)(21,24,26,28)(22,23,27,31)(25,35,34,32)(29,36,33,30);(5,8,9,10)(6,7,11,12)(13,16,17,15)(14,19,20,18)(21,25,26,34)(22,29,27,33)(23,30,31,36)(24,32,28,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=i,d^2=g,e^2=fi,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, 2960059);