#301203

order	512
nilpotency class	3
isoclinism	isoclinic class of #264427
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×#3332	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1090 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)(8,18)(9,19)(10,12)(11,16)(14,15)(17,20)(21,22)(23,31)(24,27)(25,30)(26,29)(28,34)(32,35)(33,36);(5,7,15,8,10,17,9,16)(6,11,19,20,12,18,14,13)(22,27)(23,32)(25,26)(28,31)(33,34)(35,36);(1,2,3,4)(6,12)(11,18)(13,20)(14,19)(21,23,26,32,24,33,25,34)(22,28,30,36,27,35,29,31);(5,8,15,17,10,16,9,7)(6,13,19,11,12,20,14,18)(22,27)(28,35)(29,30)(31,36)

Polycyclic presentation:

2^9|b^2=ehi,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,d^b=dh,f^b=fi,g^b=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 208399);