#235721

order	512
nilpotency class	3
isoclinism	isoclinic class of #234099
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
prime radical	C₂×#3251	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1333 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)(7,12)(8,13)(9,18)(10,14)(11,15)(16,19)(17,20)(22,26)(23,29)(24,32)(28,33)(31,34)(35,36);(1,2)(3,4)(5,7,10,16)(6,11,14,20)(8,15,9,17)(12,13,19,18)(21,22,25,28)(23,30,31,27)(24,33,32,26)(29,36,34,35);(2,4)(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,20)(14,19)(21,23)(22,27)(24,34)(25,31)(26,35)(28,30)(29,32)(33,36);(5,8,10,9)(6,13,14,18)(7,15,16,17)(11,19,20,12)(23,31)(27,30)(29,34)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142917);