#4131226

order	512
nilpotency class	2
isoclinism	isoclinic class of #4122169
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2190 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=f,d^2=fh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4038422);