#4957817

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

1968 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fgi,c^2=fgh,e^2=gh,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, 4865013);