#419894

order	512
nilpotency class	3
isoclinism	isoclinic class of #418464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60456	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1551 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^c=hi,f^d=fi,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, 327090);