#8922724

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

4125 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)(3,9,10,4)(5,13,14,8)(11,12,16,15)(17,18,22,24)(19,23,27,31)(20,25,26,21)(28,29,30,32)(33,34,38,41)(35,40,44,39)(36,45,46,37)(42,43,47,48);(2,7)(4,9)(8,13)(12,15)(17,19)(18,23)(20,30)(21,29)(22,27)(24,31)(25,32)(26,28)(33,35)(34,39)(36,42)(37,43)(38,44)(40,41)(45,48)(46,47);(2,7)(3,10)(5,14)(12,15)(18,24)(19,27)(21,25)(28,30)(36,46)(37,45)(42,47)(43,48);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,19)(18,23)(20,30)(21,29)(22,27)(24,31)(25,32)(26,28)(33,36)(34,37)(35,42)(38,46)(39,43)(40,48)(41,45)(44,47);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,20,22,26)(18,25,24,21)(19,28,27,30)(23,29,31,32)(33,35)(34,40)(36,47)(37,43)(38,44)(39,41)(42,46)(45,48);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,21,22,25)(18,26,24,20)(19,29,27,32)(23,28,31,30)(33,37)(34,36)(35,43)(38,45)(39,42)(40,47)(41,46)(44,48)

Polycyclic presentation:

2^9|a^2=gi,d^2=hi,e^2=i,f^2=h,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, 8829920);