#501034

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1001 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gh,h^c=hi,e^d=ei,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, 408230);