#4600701

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2389 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)(5,6,8,10)(7,11,12,9)(13,14,17,19)(15,16,20,18)(23,29)(24,31)(26,32)(27,34);(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,22,25,28)(23,26,29,32)(24,27,31,34)(30,33,35,36);(7,12)(9,11)(14,19)(15,20)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,23,25,29)(22,26,28,32)(24,30,31,35)(27,33,34,36);(5,7)(6,9)(8,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,24)(22,27)(23,30)(25,31)(26,33)(28,34)(29,35)(32,36)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4507897);