#5474070

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5381266);