#119890

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#388
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fi,e^c=ehi,f^c=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27086);