#8285566

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

4120 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=h,e^2=gh,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, 8192762);