#3334940

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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2934 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fg,d^2=f,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, 3242136);