#340294

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

613 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=ehi,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=de,e^b=eh,d^c=dh,f^c=fi,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, 247490);