#500680

order	512
nilpotency class	4
isoclinism	isoclinic class of #500676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1179 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=ei,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^c=fh,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, 407876);