#232740

order	512
nilpotency class	3
isoclinism	isoclinic class of #232736
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
prime radical	#60916	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1227 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)(7,12)(11,14)(13,16)(18,23)(19,27)(20,21)(25,26)(28,31)(29,30);(1,3,6,14)(2,7,9,16)(4,13,5,12)(8,11,15,10)(17,18,22,26)(19,28,30,31)(20,23,21,25)(24,29,32,27);(1,2)(3,11)(4,8)(5,15)(6,9)(7,12)(10,14)(13,16)(17,19)(18,24)(20,29)(21,27)(22,30)(23,28)(25,31)(26,32);(1,4,6,5)(2,8,9,15)(3,12,14,13)(7,10,16,11)(17,20,22,21)(18,23,26,25)(19,29,30,27)(24,28,32,31)

Polycyclic presentation:

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

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, 139936);