#6465502

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2548 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi

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, 6372698);