#499755

order	512
nilpotency class	4
isoclinism	isoclinic class of #499754
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	#61030	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1295 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406951);