#193098

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
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

866 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)(5,6,11,16)(7,17,8,19)(9,14,10,15)(12,20,13,18)(23,30)(24,25)(27,34)(28,32)(31,33)(35,36);(1,3,4,2)(5,7,11,8)(6,12,16,13)(9,20,10,18)(14,17,15,19)(21,22)(23,27)(24,28)(25,32)(26,29)(30,35)(31,36)(33,34);(5,8,11,7)(6,13,16,12)(9,20,10,18)(14,17,15,19)(21,23)(22,27)(24,30)(25,33)(26,31)(28,35)(29,36)(32,34);(5,9,11,10)(6,14,16,15)(7,18,8,20)(12,19,13,17)(21,24,26,25)(22,28,29,32)(23,30,31,33)(27,35,36,34)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100294);