#8406463

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	#57000	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3854 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,e^2=gh,f^2=h,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, 8313659);