#4414447

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 4321643);