#193029

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

940 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)(6,11)(7,15)(8,9)(13,14)(16,17)(18,19)(22,27)(24,25)(28,35)(29,32)(30,33)(34,36);(1,3,4,2)(5,6)(7,12)(8,13)(9,11)(10,14)(15,19)(16,20)(17,18)(21,22,26,29)(23,28,31,36)(24,27,25,32)(30,34,33,35);(5,7)(6,12)(8,15)(9,17)(10,16)(11,18)(13,19)(14,20)(21,23)(22,28)(24,30)(25,33)(26,31)(27,34)(29,36)(32,35);(5,8,10,9)(6,13,14,11)(7,15,16,17)(12,19,20,18)(21,24,26,25)(22,27,29,32)(23,30,31,33)(28,34,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100225);