#8114391

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	#52769	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021587);