#236178

order	512
nilpotency class	3
isoclinism	isoclinic class of #234079
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1322 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143374);