#5244353

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,6)(4,7)(9,10,15,16)(11,12,14,13)(17,18,21,19)(20,23,24,22)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40);(1,2,5,6)(3,7,8,4)(9,11)(10,13)(12,16)(14,15)(17,19,21,18)(20,23,24,22)(26,30)(27,34)(28,36)(38,39);(3,8)(4,7)(9,12,15,13)(10,11,16,14)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,13,15,12)(10,11,16,14)(17,20)(18,22)(19,23)(21,24)(25,28,29,36)(26,32,30,37)(27,35,34,40)(31,38,33,39);(1,4)(2,3)(5,7)(6,8)(9,14,15,11)(10,12,16,13)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fgi,c^2=gh,d^2=g,e^2=gh,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, 5151549);