#346967

order	512
nilpotency class	3
isoclinism	isoclinic class of #346950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62845	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2143 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:

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

Polycyclic presentation:

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

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, 254163);