#419890

order	512
nilpotency class	3
isoclinism	isoclinic class of #418466
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	#60454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1629 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)(7,13)(8,20,22,26)(9,12)(10,15)(11,16)(14,23,30,21)(17,31,33,34)(18,25)(19,29)(24,28)(27,32,36,35);(1,2)(3,4)(6,12)(8,21)(9,10)(13,25)(15,16)(17,32)(18,24)(20,26)(22,23)(28,29)(31,34)(33,35);(2,4)(5,7,11,19)(6,13,16,29)(8,17,22,33)(9,18,10,24)(12,25,15,28)(14,27,30,36)(20,34,26,31)(21,32,23,35);(5,8,9,21,11,22,10,23)(6,14,15,20,16,30,12,26)(7,17,18,32,19,33,24,35)(13,27,28,34,29,36,25,31)

Polycyclic presentation:

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

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, 327086);