#2490770

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	#16258	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2225 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=fh,d^2=g,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2397966);