#233688

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,19)(18,20)(22,26)(24,31)(27,35)(28,32)(30,33)(34,36);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,15)(13,20)(14,19)(21,22)(23,27)(24,32)(25,28)(26,31)(29,36)(30,34)(33,35);(2,4)(5,7)(6,11)(8,17)(9,16)(10,18)(12,20)(13,15)(14,19)(21,23)(22,27)(24,33)(25,29)(26,34)(28,36)(30,31)(32,35);(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,15,19,20)(23,29)(27,36)(30,33)(34,35)

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,c^b=cg,e^b=eh,f^b=fi,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, 140884);