#4928854

order	512
nilpotency class	2
isoclinism	isoclinic class of #4928752
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32);(9,11,14,12)(10,15,13,16)(18,19)(20,27)(24,26)(28,32);(5,6,7,8)(9,12,14,11)(10,15,13,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31);(1,2)(3,4)(5,6)(7,8)(9,13,14,10)(11,15,12,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31);(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)

Polycyclic presentation:

2^9|b^2=fh,c^2=fgh,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4836050);