#3300282

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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

1676 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fi,d^2=fgi,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3207478);