#8406813

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	#40866	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4111 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)(3,9)(4,12)(5,6)(7,11)(8,14)(10,15)(13,16)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(33,34,39,45)(35,44,47,40)(36,43,38,41)(37,42,48,46);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,19,23,28)(18,24,27,32)(20,31,25,22)(21,26,30,29)(33,35)(34,40)(36,46)(37,41)(38,42)(39,47)(43,48)(44,45);(2,6)(7,9)(8,12)(15,16)(19,28)(22,31)(24,32)(26,29)(33,36,39,38)(34,41,45,43)(35,46,47,42)(37,44,48,40);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20)(18,21)(19,22)(23,25)(24,29)(26,32)(27,30)(28,31)(33,37)(34,42)(35,41)(36,44)(38,40)(39,48)(43,47)(45,46);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,21)(18,25)(19,29)(20,27)(22,32)(23,30)(24,31)(26,28)(33,38,39,36)(34,43,45,41)(35,46,47,42)(37,44,48,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,22)(18,26)(19,20)(21,32)(23,31)(24,30)(25,28)(27,29)(33,35)(34,44)(36,42)(37,41)(38,46)(39,47)(40,45)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=gh,e^2=g,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, 8314009);