#8831271

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	#48409	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,e^2=g,f^2=h,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, 8738467);