#111172

order	512
nilpotency class	3
isoclinism	isoclinic class of #111151
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

586 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di

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, 18368);