#411704

order	512
nilpotency class	3
isoclinism	isoclinic class of #396373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1840 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:

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

Polycyclic presentation:

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

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, 318900);