#2607958

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 2097152
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

1882 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=h,c^2=g,d^2=f,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, 2515154);