#194418

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60790	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1107 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,d^b=dh,e^b=eh,f^b=fi,g^b=gh,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, 101614);