#2610304

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2598 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)(5,6)(7,9)(8,11)(10,12)(14,18)(15,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,35)(32,36);(6,7)(10,11)(13,14)(15,19)(16,20)(17,18)(21,22,25,28)(23,26,30,34)(24,27,32,36)(29,33,31,35);(5,6,9,7)(8,10,12,11)(13,15,17,20)(14,19,18,16)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,35)(32,36);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,16)(14,15)(17,19)(18,20)(21,23,25,30)(22,26,28,34)(24,31,32,29)(27,35,36,33);(5,8)(6,10)(7,11)(9,12)(13,14)(15,19)(16,20)(17,18)(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=fg,c^2=g,d^2=h,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, 2517500);