#464181

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

1799 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)(16,30)(23,34)(29,36)(31,33);(1,2,4,3)(5,6,9,13,10,14,22,28)(7,15,17,24,18,27,21,11)(8,12,20,26,19,25,32,35)(16,29,31,34,30,36,33,23);(2,3)(5,7)(6,11)(8,16)(9,21)(10,18)(12,23)(13,27)(14,24)(15,28)(17,22)(19,30)(20,33)(25,34)(26,36)(29,35)(31,32);(1,3,4,2)(5,8)(6,12)(7,16)(9,20)(10,19)(11,23)(13,26)(14,25)(15,29)(17,31)(18,30)(21,33)(22,32)(24,34)(27,36)(28,35)

Polycyclic presentation:

2^9|b^2=g,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,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, 371377);