#3943049

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2123 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fhi,c^2=i,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, 3850245);