#845282

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

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

1920 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)(3,7)(4,5)(6,8)(17,18)(19,21)(20,22)(23,24)(25,26,28,29)(27,31,32,30);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,20,23)(18,21,22,24)(26,29)(27,32);(1,3)(2,6)(4,8)(5,7)(10,13)(14,16)(27,32)(30,31);(3,8)(6,7)(18,22)(21,24)(25,27,28,32)(26,30,29,31);(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgi,d^2=fg,e^2=h,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, 752478);