#4456852

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 2097152
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

1578 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=fgi,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,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, 4364048);