#865726

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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

1600 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=gh,c^2=fgi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 772922);