#194411

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
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	#60782	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1099 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:

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

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=di,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, 101607);