#502188

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#576	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

744 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:

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

Polycyclic presentation:

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

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, 409384);