#193298

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

944 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)(7,10)(8,11)(9,13)(12,16)(14,18)(15,17)(19,20)(21,22)(23,32)(24,35)(25,31)(26,30)(27,29)(28,34)(33,36);(1,3,4,2)(5,7)(6,10)(8,15)(9,14)(11,17)(12,19)(13,18)(16,20)(21,23)(22,28)(24,29)(25,35)(26,34)(27,33)(30,36)(31,32);(6,11)(8,16)(10,17)(12,13)(15,20)(18,19)(21,24,27,28)(22,23,29,33)(25,32,26,36)(30,35,31,34);(7,14)(10,18)(15,20)(17,19)(22,29)(23,33)(30,31)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100494);