#8839196

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	#49403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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)(3,7)(4,8)(5,14)(6,10)(9,15)(11,12)(13,16)(17,18)(19,20)(21,24)(22,30)(23,26)(25,32)(27,29)(28,31)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,3)(2,7)(4,5)(6,11)(8,14)(9,16)(10,12)(13,15)(17,19,23,24)(18,20,26,21)(22,31,32,27)(25,29,30,28)(33,35,37,43)(34,38,40,41)(36,44,45,42)(39,47,48,46);(3,11)(4,13)(7,12)(8,16)(17,20,23,21)(18,24,26,19)(22,28,32,29)(25,31,30,27)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,4,6,13)(2,8,10,16)(3,5,11,15)(7,14,12,9)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48);(1,5)(2,9)(3,4)(6,15)(7,16)(8,12)(10,14)(11,13)(17,21,23,20)(18,19,26,24)(22,29,32,28)(25,27,30,31)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,2)(3,12)(4,8)(5,9)(6,10)(7,11)(13,16)(14,15)(17,22,23,32)(18,25,26,30)(19,27,24,31)(20,28,21,29)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8746392);