#5279828

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2198 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)(14,18)(15,16)(21,22,26,27)(23,28,31,36)(24,25,30,29)(32,33,34,35);(5,6,8,9)(7,11,12,10)(13,14,17,18)(15,19,16,20)(22,27)(23,31)(24,30)(33,35);(6,9)(10,11)(13,15,17,16)(14,19,18,20)(21,23)(22,28)(24,34)(25,35)(26,31)(27,36)(29,33)(30,32);(1,2,3,4)(5,7,8,12)(6,10,9,11)(13,16,17,15)(14,19,18,20)(21,24,26,30)(22,29,27,25)(23,32,31,34)(28,35,36,33);(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,19)(17,18)(21,25,26,29)(22,30,27,24)(23,33,31,35)(28,34,36,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fh,d^2=i,e^2=fg,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, 5187024);