#8322187

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4222 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=g,e^2=gh,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, 8229383);