#8924739

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#48667	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8831935);