#1388041

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2133 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)(7,11)(10,12)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(13,14,17,16)(15,19,20,18)(21,22,25,28)(23,26,31,29)(24,27,33,36)(30,34,32,35);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,15,17,20)(14,18,16,19)(21,22)(23,26)(24,27)(25,28)(29,31)(30,34)(32,35)(33,36);(5,7)(6,10)(8,12)(9,11)(21,23,25,31)(22,26,28,29)(24,32,33,30)(27,35,36,34);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,16,17,14)(15,19,20,18)(21,24,25,33)(22,27,28,36)(23,30,31,32)(26,34,29,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1295237);