#119804

order	512
nilpotency class	3
isoclinism	isoclinic class of #119801
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

531 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:

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

Polycyclic presentation:

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

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, 27000);