#300689

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1033 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207885);