#2241616

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
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	#14722	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2321 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(18,21)(19,23)(26,29)(30,32);(1,3)(2,4)(5,6)(7,8)(9,10,12,14)(11,13,16,15)(19,23)(22,24)(25,26,28,29)(27,30,31,32);(3,6)(4,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,21)(23,24)(25,27,28,31)(26,30,29,32);(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32);(1,4,5,8)(2,6,7,3)(9,11,12,16)(10,13,14,15)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,31)(29,32)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fh,c^2=gi,e^2=h,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, 2148812);