#5480701

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2243 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)(7,11)(8,9)(10,12)(14,17)(18,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(5,6)(7,11)(8,9)(10,12)(15,19)(18,20)(21,22,25,28)(23,26,31,29)(24,32,33,27)(30,35,36,34);(6,9)(10,11)(13,14,16,17)(15,18,19,20)(21,22)(23,26)(24,32)(25,28)(27,33)(29,31)(30,35)(34,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23,25,31)(22,26,28,29)(24,30,33,36)(27,34,32,35);(5,7)(6,10)(8,12)(9,11)(13,15,16,19)(14,18,17,20)(21,24)(22,27)(23,30)(25,33)(26,34)(28,32)(29,35)(31,36)

Polycyclic presentation:

2^9|b^2=f,c^2=fgh,d^2=f,e^2=fgh,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, 5387897);