#423173

order	512
nilpotency class	3
isoclinism	isoclinic class of #419839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1327 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:

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

Polycyclic presentation:

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

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, 330369);