#249845

order	512
nilpotency class	3
isoclinism	isoclinic class of #249839
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1012 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:

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

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,g^c=gh,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, 157041);