#1367479

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2560 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)(13,14)(15,19)(16,18)(17,20)(21,22,26,27)(23,28,30,25)(24,29,33,35)(31,32,36,34);(5,6,9,8)(7,10,11,12)(13,15,18,17)(14,19,16,20)(22,27)(23,30)(29,35)(31,36);(1,2)(3,4)(13,16,18,14)(15,19,17,20)(21,23)(22,28)(24,31)(25,27)(26,30)(29,32)(33,36)(34,35);(5,7)(6,10)(8,12)(9,11)(21,24)(22,29)(23,31)(25,34)(26,33)(27,35)(28,32)(30,36);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,17,18,15)(14,19,16,20)(21,25,26,28)(22,23,27,30)(24,32,33,34)(29,36,35,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=fhi,c^2=fh,e^2=hi,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, 1274675);