#232693

order	512
nilpotency class	3
isoclinism	isoclinic class of #232688
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

1271 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gh,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139889);