#3335454

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

2822 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)(9,10,12,13)(11,14,15,16)(18,22)(19,25)(20,27)(31,32);(5,6,7,8)(10,13)(14,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32);(5,6)(7,8)(11,15)(14,16)(17,19,21,25)(18,23,22,30)(20,28,27,26)(24,32,29,31);(1,2)(3,4)(9,10)(11,16)(12,13)(14,15)(17,18)(19,23)(20,29)(21,22)(24,27)(25,30)(26,32)(28,31);(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,31)(25,28)(30,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=g,e^2=fg,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, 3242650);