#411853

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

2108 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)(5,6,12,16)(7,11,15,10)(8,17,23,29)(9,24,27,18)(13,21,19,22)(14,26,30,25)(20,28,35,32)(31,34,36,33);(1,2,3,4)(5,7,10,6,12,15,11,16)(8,19,21,17,23,13,22,29)(9,18,25,14,27,24,26,30)(20,32,33,31,35,28,34,36);(5,8)(6,13)(7,17)(9,20)(10,21)(11,22)(12,23)(14,28)(15,29)(16,19)(18,31)(24,36)(25,33)(26,34)(27,35)(30,32);(5,9,11,26,12,27,10,25)(6,14,7,18,16,30,15,24)(8,20,22,34,23,35,21,33)(13,28,17,31,19,32,29,36)

Polycyclic presentation:

2^9|a^2=i,b^2=e,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,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, 319049);