#499022

order	512
nilpotency class	4
isoclinism	isoclinic class of #498812
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1072 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406218);