#209571

order	512
nilpotency class	3
isoclinism	isoclinic class of #18252
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	#3455	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

806 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:

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

Polycyclic presentation:

2^9|a^2=g,c^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,d^c=dhi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116767);