#4132977

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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

2464 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)(9,10,13,15)(11,12,16,14)(17,18,22,23)(19,27,30,24)(20,25,26,21)(28,29,31,32);(5,6,7,8)(9,10,13,15)(11,12,16,14)(18,23)(21,25)(24,27)(29,32);(5,6)(7,8)(9,11,13,16)(10,14,15,12)(17,19)(18,24)(20,31)(21,32)(22,30)(23,27)(25,29)(26,28);(1,2)(3,4)(9,11)(10,12)(13,16)(14,15)(17,20)(18,25)(19,28)(21,23)(22,26)(24,32)(27,29)(30,31);(9,12)(10,11)(13,14)(15,16)(17,21,22,25)(18,26,23,20)(19,29,30,32)(24,31,27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4040173);