#5244371

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

1427 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=fgh,d^2=g,e^2=fgh,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, 5151567);