#120951

order	512
nilpotency class	3
isoclinism	isoclinic class of #10386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

426 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28147);