#198706

order	512
nilpotency class	3
isoclinism	isoclinic class of #198534
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1013 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:

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

Polycyclic presentation:

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

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, 105902);