#3089915

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2053 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,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, 2997111);