#8746075

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	#52663	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4012 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gi,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8653271);