#3379930

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ghi,c^2=fh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3287126);