#4142244

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

2457 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)(5,6,9,7)(8,11,12,10)(13,14,16,18)(15,19,20,17)(21,22)(23,26)(24,29)(25,30)(27,32)(28,33)(31,35)(34,36);(5,6,9,7)(8,11,12,10)(13,14,16,18)(15,17,20,19)(22,23)(25,33)(27,29)(31,34);(6,7)(10,11)(13,15,16,20)(14,17,18,19)(21,23,26,22)(24,27,32,29)(25,28,33,30)(31,36,34,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,17)(16,20)(18,19)(21,24,26,32)(22,27,23,29)(25,34,33,31)(28,36,30,35);(5,8)(6,10)(7,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,25)(22,28)(23,30)(24,31)(26,33)(27,35)(29,36)(32,34)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fgh,d^2=h,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, 4049440);