#420457

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1755 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)(7,17)(8,22,25,26)(9,12)(10,15)(11,16)(13,21)(14,24,30,23)(18,28)(19,33,32,27)(20,29)(31,35,36,34);(1,2)(3,4)(6,12)(7,18)(8,23)(9,10)(13,17)(15,16)(19,34)(22,26)(24,25)(28,29)(31,36)(32,35);(2,4)(5,7,9,20,11,18,10,21)(6,13,15,28,16,29,12,17)(8,19,23,35,25,32,24,34)(14,27,22,36,30,33,26,31);(5,8,9,23,11,25,10,24)(6,14,15,22,16,30,12,26)(7,19,20,35,18,32,21,34)(13,27,28,36,29,33,17,31)

Polycyclic presentation:

2^9|a^2=ef,c^2=e,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,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, 327653);