#3943273

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1979 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=i,d^2=fg,e^2=f,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, 3850469);