#502189

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	#33522	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

752 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)(5,6,10,16)(7,18,22,14)(8,23,24,35)(9,13,27,31)(11,17,12,28)(15,20,32,25)(19,34,26,33)(21,36,29,30);(1,3,4,2)(6,13)(7,19)(9,25)(11,12)(14,23)(15,17)(16,31)(18,35)(20,27)(21,29)(22,26)(28,32)(30,33)(34,36);(2,3)(5,7)(6,14)(8,20)(9,21)(10,22)(11,26)(12,19)(13,30)(15,23)(16,18)(17,34)(24,25)(27,29)(28,33)(31,36)(32,35);(5,8)(6,14)(7,20)(9,26)(10,24)(11,21)(12,29)(13,23)(15,30)(16,18)(17,33)(19,27)(22,25)(28,34)(31,35)(32,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,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, 409385);