#4951395

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 4858591);