#206471

order	512
nilpotency class	3
isoclinism	isoclinic class of #19697
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

916 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,15)(14,16)(17,18,22,26,23,27,21,36)(19,24,31,39,29,38,30,45)(20,32,34,40,35,46,33,25)(28,41,44,47,42,48,43,37);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15)(17,19,22,31,23,29,21,30)(18,24,26,39,27,38,36,45)(20,28,34,44,35,42,33,43)(25,37,32,41,40,47,46,48);(2,7)(3,10)(8,12)(11,16)(17,20,21,33,23,35,22,34)(18,25,26,32,27,40,36,46)(19,28,30,43,29,42,31,44)(24,37,39,41,38,47,45,48);(19,29)(24,38)(28,42)(30,31)(37,47)(39,45)(41,48)(43,44)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=di

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, 113667);