#8323313

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 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)(3,11,12,8)(4,13,15,9)(5,10,16,14)(17,18,23,26)(19,22,25,31)(20,21,24,30)(27,28,29,32)(34,38)(36,43)(39,44)(42,47);(2,7)(3,12)(9,13)(10,14)(17,19)(18,22)(20,27)(21,28)(23,25)(24,29)(26,31)(30,32)(35,41)(39,44)(42,47)(46,48);(2,7)(5,16)(8,11)(9,13)(19,25)(22,31)(27,29)(28,32)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,3)(2,8)(4,14)(5,9)(6,12)(7,11)(10,15)(13,16)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,30)(31,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,4,6,15)(2,9,7,13)(3,10,12,14)(5,8,16,11)(17,21)(18,24)(19,28)(20,26)(22,29)(23,30)(25,32)(27,31)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48);(1,5,6,16)(2,10,7,14)(3,9,12,13)(4,8,15,11)(17,22,23,31)(18,25,26,19)(20,28,24,32)(21,29,30,27)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8230509);