#6404492

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1779 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fg,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6311688);