#6498413

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469944
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi

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, 6405609);