#3299346

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

(2,5)(6,8)(9,10,13,12)(11,14,16,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(9,10,13,12)(11,15,16,14)(18,22)(20,28)(23,25)(26,29)(36,44)(39,46)(42,48)(45,47);(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,28)(30,31)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,2,4,5)(3,6,7,8)(9,10,13,12)(11,14,16,15)(17,20,21,28)(18,24,22,30)(19,26,27,29)(23,31,25,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,18)(19,23)(20,30)(21,22)(24,28)(25,27)(26,32)(29,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=fgh,d^2=ghi,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3206542);