#8908452

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

4261 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)(3,10)(4,9)(5,14)(6,7)(8,11)(12,16)(13,15)(17,18)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32)(33,34)(35,41)(36,37)(38,42)(39,45)(40,46)(43,48)(44,47);(2,7)(4,14)(8,10)(12,15)(17,19,23,20)(18,24,25,21)(22,27,31,28)(26,32,30,29)(33,35)(34,39)(36,43)(37,44)(38,45)(40,48)(41,42)(46,47);(2,7)(3,11)(5,9)(12,15)(18,25)(19,20)(22,31)(29,32)(36,46)(37,40)(43,47)(44,48);(1,3,6,11)(2,8,7,10)(4,15,14,12)(5,13,9,16)(17,20,23,19)(18,21,25,24)(22,28,31,27)(26,29,30,32)(33,36)(34,40)(35,43)(37,42)(38,46)(39,48)(41,44)(45,47);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,16)(11,15)(17,21)(18,20)(19,25)(22,29)(23,24)(26,28)(27,30)(31,32)(33,35)(34,41)(36,47)(37,44)(38,45)(39,42)(40,48)(43,46);(1,5,6,9)(2,4,7,14)(3,13,11,16)(8,15,10,12)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)(33,37,38,40)(34,36,42,46)(35,44,45,48)(39,43,41,47)

Polycyclic presentation:

2^9|b^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815648);