#4423083

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
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

1810 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=fg,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4330279);