#8418252

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	#42539	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(19,20)(21,29)(24,26)(25,30)(33,34,39,43)(35,41,45,38)(36,40,37,42)(44,46,47,48);(1,3)(2,8)(4,6)(5,15)(7,12)(9,13)(10,14)(11,16)(17,18,23,22)(19,26,20,24)(21,30,29,25)(27,31,28,32)(33,35,39,45)(34,38,43,41)(36,44,37,47)(40,48,42,46);(3,12)(6,13)(8,16)(10,15)(17,19,23,20)(18,24,22,26)(21,27,29,28)(25,31,30,32)(33,36,39,37)(34,40,43,42)(35,44,45,47)(38,48,41,46);(1,4)(2,5)(3,13)(6,12)(7,9)(8,10)(11,14)(15,16)(17,20,23,19)(18,24,22,26)(21,27,29,28)(25,32,30,31)(33,35)(34,41)(36,44)(37,47)(38,43)(39,45)(40,46)(42,48);(1,5)(2,9)(3,10)(4,11)(6,8)(7,14)(12,15)(13,16)(17,21,23,29)(18,25,22,30)(19,27,20,28)(24,31,26,32)(33,37,39,36)(34,42,43,40)(35,44,45,47)(38,48,41,46);(1,6,7,13)(2,10,11,15)(3,4,12,9)(5,8,14,16)(17,22,23,18)(19,24,20,26)(21,30,29,25)(27,31,28,32)(33,38,39,41)(34,35,43,45)(36,46,37,48)(40,47,42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8325448);