#192268

order	512
nilpotency class	3
isoclinism	isoclinic class of #18192
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	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

849 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99464);