#203174

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

627 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dhi,d^c=dh,f^c=fhi

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, 110370);