#4198971

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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

1386 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fhi,c^2=fh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,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, 4106167);