#322054

order	512
nilpotency class	3
isoclinism	isoclinic class of #288715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

753 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)(7,12)(8,17)(9,19)(10,14)(11,15)(13,18)(16,20)(21,22,25,28,26,29,30,23)(24,31,34,32,35,36,33,27);(5,7,9,15,10,16,18,8)(6,11,13,12,14,17,19,20)(21,23,30,29,26,28,25,22)(24,32,33,31,35,27,34,36);(2,4)(21,24,26,35)(22,27,29,32)(23,31,28,36)(25,33,30,34);(5,8,18,16,10,15,9,7)(6,12,19,11,14,20,13,17)(21,25,26,30)(22,28,29,23)(24,33,35,34)(27,36,32,31)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^c=fi,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229250);