#411851

order	512
nilpotency class	3
isoclinism	isoclinic class of #395246
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2044 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=h,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,h^a=hi,c^b=cg,g^b=gi,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, 319047);