#5955301

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
automorphism	Aut(G) order 1048576
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1847 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fghi,c^2=fghi,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5862497);