#203892

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

668 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)(7,14)(8,10)(9,13)(11,15)(12,17)(16,19)(18,20)(23,29)(24,31)(26,32)(28,34)(30,35)(33,36);(1,3,4,2)(6,10)(8,17)(11,18)(12,13)(14,19)(16,20)(21,22)(23,26)(24,28)(25,27)(29,32)(30,33)(31,34)(35,36);(5,7)(6,11)(8,16)(9,15)(10,18)(12,19)(13,14)(17,20)(21,23,24,29,25,30,31,35)(22,26,28,32,27,33,34,36);(7,15)(11,14)(16,20)(18,19)(22,27)(23,30)(28,34)(29,35)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,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, 111088);