#8922917

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

4183 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)(19,24)(20,25)(21,26)(22,23)(27,30)(28,32)(29,31)(33,34,38,37)(35,39,41,43)(36,40,44,45)(42,46,47,48);(2,7)(4,9)(8,13)(12,15)(17,19,23,27)(18,24,22,30)(20,28,29,21)(25,32,31,26)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(2,7)(3,10)(5,14)(12,15)(18,22)(19,27)(20,29)(26,32)(35,41)(36,44)(39,43)(40,45);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,20)(18,25)(19,28)(21,27)(22,31)(23,29)(24,32)(26,30)(33,36,38,44)(34,40,37,45)(35,42,41,47)(39,46,43,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21,23,28)(18,26,22,32)(19,29,27,20)(24,31,30,25)(33,37,38,34)(35,43,41,39)(36,40,44,45)(42,46,47,48);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,22,23,18)(19,30,27,24)(20,31,29,25)(21,32,28,26)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,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, 8830113);