#198105

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1053 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,d^c=dh,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105301);