#1294941

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2244 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^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, 1202137);