#236192

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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1396 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,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, 143388);