#194390

order	512
nilpotency class	3
isoclinism	isoclinic class of #18562
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

1117 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101586);