#4963781

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1976 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=fgi,d^2=fg,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, 4870977);