#464178

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=g,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371374);