#340038

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

634 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247234);