#499737

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61056	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1292 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,e^b=eh,h^b=hi,f^c=fi,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, 406933);