#188999

order	512
nilpotency class	3
isoclinism	isoclinic class of #188956
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
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

829 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^c=dh,f^c=fh,g^c=ghi

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, 96195);