#2633480

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2700 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,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, 2540676);