#120434

order	512
nilpotency class	3
isoclinism	isoclinic class of #120431
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27630);