#4376128

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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

1433 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=f,d^2=fhi,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4283324);