#198468

order	512
nilpotency class	3
isoclinism	isoclinic class of #198162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1139 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)(3,7,11,16)(4,8)(5,12)(6,9)(10,14,13,15)(17,18,22,27,23,28,21,26)(19,29,32,40,33,39,31,24)(20,34,35,25,37,41,36,42)(30,38,44,47,46,48,45,43);(1,3)(2,7,9,16)(4,10)(5,13)(6,11)(8,14,12,15)(17,19)(18,24,28,40)(20,30)(21,31)(22,32)(23,33)(25,38,42,48)(26,39,27,29)(34,43,41,47)(35,44)(36,45)(37,46);(1,4,6,5)(2,8,9,12)(3,10,11,13)(7,14,16,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,41)(27,34)(28,42)(29,43)(31,44)(32,45)(33,46)(39,47)(40,48);(1,5,6,4)(2,8,9,12)(3,10,11,13)(7,15,16,14)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,39,40,29)(25,41,42,34)(30,44,46,45)(38,47,48,43)

Polycyclic presentation:

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

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, 105664);