#346171

order	512
nilpotency class	3
isoclinism	isoclinic class of #346096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

478 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=ei,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=defi,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253367);