#246191

order	512
nilpotency class	3
isoclinism	isoclinic class of #246159
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	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153387);