#194618

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
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

1072 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,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, 101814);