#512400

order	512
nilpotency class	4
isoclinism	isoclinic class of #511963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=eh,c^2=f,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,g^d=gi

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, 419596);