#192529

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

811 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:

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

Polycyclic presentation:

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

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, 99725);