#2610035

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2531 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=f,d^2=gh,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, 2517231);