#209063

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

820 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)(17,18,22,27,23,21,26,35)(19,24,31,38,30,29,37,44)(20,25,34,40,33,32,39,45)(28,41,43,48,42,36,46,47);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,36)(26,37)(27,38)(32,41)(33,42)(34,43)(35,44)(39,46)(40,47)(45,48);(17,20)(18,25)(19,28)(21,32)(22,34)(23,33)(24,36)(26,39)(27,40)(29,41)(30,42)(31,43)(35,45)(37,46)(38,47)(44,48);(17,21,22,35,23,18,26,27)(19,29,31,44,30,24,37,38)(20,32,34,45,33,25,39,40)(28,41,43,48,42,36,46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116259);