#5282278

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

1951 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)(7,9)(8,10)(11,12)(13,14,17,19)(15,16,20,18)(21,22,26,29)(23,25,30,27)(24,32,35,28)(31,36,33,34);(8,12)(10,11)(13,15,17,20)(14,18,19,16)(23,30)(25,27)(31,33)(34,36);(6,7)(10,11)(13,15,17,20)(14,16,19,18)(21,23)(22,27)(24,33)(25,29)(26,30)(28,34)(31,35)(32,36);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,16)(17,20)(18,19)(21,24,26,35)(22,28,29,32)(23,31,30,33)(25,34,27,36);(5,8)(6,10)(7,11)(9,12)(13,16)(14,15)(17,18)(19,20)(21,25,26,27)(22,23,29,30)(24,34,35,36)(28,31,32,33)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=f,d^2=i,e^2=h,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, 5189474);