#4145222

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 1048576
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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=gh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,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, 4052418);