#5160579

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2047 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,21,20,24)(25,26)(27,32)(28,30)(29,31);(4,8)(6,7)(17,19,22,20)(18,21,23,24)(25,27)(26,31)(28,32)(29,30);(2,3)(6,7)(18,23)(19,20)(25,26,30,28)(27,31,29,32);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20,22,19)(18,21,23,24)(25,28,30,26)(27,31,29,32);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,21)(18,19)(20,23)(22,24)(25,29,30,27)(26,31,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5067775);