#631778

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2384 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)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,22)(21,24)(25,26)(27,31)(28,30)(29,32);(1,2,4,6)(3,5,8,7)(17,18,20,22)(19,21,24,23)(26,28)(27,32);(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27,30,32)(26,31,28,29);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,28,30,26)(27,29,32,31);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,21)(20,24)(22,23)(25,29)(26,27)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=gi,d^2=gi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 538974);