#1487647

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=i,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1394843);