#845210

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2222 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)(3,5)(4,6)(7,8)(17,18)(19,21)(20,22)(23,24)(25,26)(27,29)(28,30)(31,32);(1,3)(2,5)(4,7)(6,8)(9,10,12,14)(11,15,16,13)(17,19,20,23)(18,21,22,24)(26,27)(30,31);(2,6)(5,8)(9,11,12,16)(10,13,14,15)(25,27,29,26)(28,31,32,30);(3,7)(5,8)(18,22)(21,24)(26,27)(30,31);(25,28,29,32)(26,30,27,31)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dfg,d^c=df

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, 752406);