#4650584

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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	#15273	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2179 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4557780);