#203896

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

642 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,e^b=eh,d^c=dhi

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, 111092);