#5213133

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2554 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)(7,9)(8,10)(11,12)(13,14,17,15)(16,19,20,18)(21,22,26,23)(24,29,31,27)(25,28,33,30)(32,35,36,34);(8,12)(10,11)(13,14,17,15)(16,19,20,18)(22,23)(24,31)(28,30)(32,36);(6,7)(10,11)(14,15)(18,19)(21,23,26,22)(24,27,31,29)(25,28,33,30)(32,35,36,34);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18)(21,24,26,31)(22,27,23,29)(25,32,33,36)(28,34,30,35);(5,8)(6,10)(7,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,25,26,33)(22,28,23,30)(24,32,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,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^b=df,e^b=eg,d^c=dfg,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, 5120329);