#4133761

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 1048576
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	C₂×#2678	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3168 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)(10,12)(11,15)(17,18)(19,20)(21,29)(22,23)(24,25)(26,30)(27,32)(28,31);(5,6,7,8)(10,12)(14,16)(18,23)(19,25)(21,30)(28,32);(5,6)(7,8)(9,10,13,12)(11,16,15,14)(17,19)(18,24)(20,23)(21,31)(22,25)(26,28)(27,30)(29,32);(1,2)(3,4)(9,11,13,15)(10,14,12,16)(17,20,22,24)(18,25,23,19)(21,32,30,28)(26,27,29,31);(9,12,13,10)(11,16,15,14)(17,21,22,30)(18,26,23,29)(19,27,25,31)(20,28,24,32)

Polycyclic presentation:

2^9|b^2=gh,c^2=fg,d^2=g,e^2=g,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, 4040957);