#209093

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	C₄×#120
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

798 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)(3,9,10,16)(4,12,14,7)(6,13,15,11)(17,18,22,21,23,26,35,34)(19,24,30,28,29,37,44,43)(20,25,33,31,32,38,46,45)(27,39,42,48,41,36,47,40);(1,3,5,10)(2,6,8,15)(4,13,14,11)(7,16,12,9)(17,19)(18,24)(20,27)(21,28)(22,30)(23,29)(25,36)(26,37)(31,40)(32,41)(33,42)(34,43)(35,44)(38,39)(45,48)(46,47);(17,20)(18,25)(19,27)(21,31)(22,33)(23,32)(24,36)(26,38)(28,40)(29,41)(30,42)(34,45)(35,46)(37,39)(43,48)(44,47);(3,10)(6,15)(9,16)(11,13)(17,21,22,26,23,34,35,18)(19,28,30,37,29,43,44,24)(20,31,33,38,32,45,46,25)(27,40,42,39,41,48,47,36)

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,d^a=di,e^a=eh,d^b=dh,e^b=eh,f^b=fi,e^c=ei

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, 116289);