#201174

order	512
nilpotency class	3
isoclinism	isoclinic class of #201112
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,8,5,7,9,15)(3,6,11,13,10,14,12,16)(17,18,22,27,23,28,21,26)(19,29,31,39,33,41,32,24)(20,34,35,42,37,25,36,40)(30,43,46,38,45,48,44,47);(1,3,9,11,5,10,4,12)(2,6,8,16,7,14,15,13)(17,19,23,33)(18,24)(20,30,37,45)(21,32,22,31)(25,38)(26,41)(27,29)(28,39)(34,47)(35,46,36,44)(40,48)(42,43);(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,40)(27,42)(28,34)(29,43)(31,44)(32,46)(33,45)(39,47)(41,48);(2,7)(3,10)(8,15)(11,12)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,39,41)(25,40,34,42)(30,44,45,46)(38,43,47,48)

Polycyclic presentation:

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

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, 108370);