#3614909

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	#16249	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2709 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=gh,e^2=fghi,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, 3522105);