#8406666

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	#40730	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4310 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)(3,11)(4,9)(5,15)(6,8)(7,14)(10,12)(13,16)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(33,34,38,37)(35,41,43,39)(36,40,45,46)(42,48,44,47);(1,3,6,14)(2,7,8,11)(4,12,13,5)(9,15,16,10)(17,19,23,28)(18,24,27,32)(20,31,25,22)(21,26,30,29)(33,35)(34,39)(36,44)(37,41)(38,43)(40,48)(42,45)(46,47);(2,8)(7,11)(9,16)(10,15)(19,28)(22,31)(24,32)(26,29)(33,34,38,37)(35,39,43,41)(36,40,45,46)(42,47,44,48);(1,4)(2,9)(3,12)(5,14)(6,13)(7,15)(8,16)(10,11)(17,20)(18,21)(19,22)(23,25)(24,29)(26,32)(27,30)(28,31)(33,36)(34,40)(35,42)(37,46)(38,45)(39,47)(41,48)(43,44);(1,5,6,12)(2,10,8,15)(3,13,14,4)(7,16,11,9)(17,21)(18,25)(19,29)(20,27)(22,32)(23,30)(24,31)(26,28)(33,37,38,34)(35,39,43,41)(36,40,45,46)(42,48,44,47);(1,2)(3,7)(4,9)(5,10)(6,8)(11,14)(12,15)(13,16)(17,22)(18,26)(19,20)(21,32)(23,31)(24,30)(25,28)(27,29)(33,35)(34,41)(36,42)(37,39)(38,43)(40,48)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^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, 8313862);