#4452879

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 16777216
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=g,d^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, 4360075);