#3337488

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2726 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=gh,e^2=g,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, 3244684);