#346983

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2069 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=i,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, 254179);