#235395

order	512
nilpotency class	3
isoclinism	isoclinic class of #235381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1254 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=di,e^b=eh,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142591);