#247036

order	512
nilpotency class	3
isoclinism	isoclinic class of #247017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18332	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1067 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

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, 154232);