#197991

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105187);