#3547137

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1970 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(18,20)(19,23)(27,31)(30,32);(2,5)(6,8)(9,10,12,14)(11,13,16,15)(17,18,21,20)(19,22,23,24)(26,29)(30,32);(3,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,26,28,29)(27,30,31,32);(1,2)(3,8)(4,5)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,27,28,31)(26,30,29,32);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,21,18)(19,22,23,24)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3454333);