#3943033

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2218 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=i,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, 3850229);