#4033835

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2708 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,c^2=fghi,d^2=fghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3941031);