#5475122

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=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, 5382318);