#3537205

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2496 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=fhi,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, 3444401);