#4198612

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fghi,c^2=fgh,d^2=h,e^2=f,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, 4105808);