#5021377

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
automorphism	Aut(G) order 8388608
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	C₂²×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2510 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,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, 4928573);