#4933391

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2223 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)(8,10,12,11)(13,14,17,15)(16,19,20,18)(21,22)(23,29)(24,32)(25,31)(26,27)(28,34)(30,33)(35,36);(13,14,17,15)(16,18,20,19)(22,27)(25,29)(28,32)(33,35);(6,7)(10,11)(14,15)(18,19)(21,23)(22,25)(24,30)(26,31)(27,29)(28,33)(32,35)(34,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18)(21,24,26,34)(22,28,27,32)(23,30,31,36)(25,33,29,35);(5,8,9,12)(6,10,7,11)(13,16,17,20)(14,18,15,19)(21,25)(22,23)(24,33)(26,29)(27,31)(28,30)(32,36)(34,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4840587);