#233278

order	512
nilpotency class	3
isoclinism	isoclinic class of #233265
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1222 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:

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

Polycyclic presentation:

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

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, 140474);