#8839134

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

4026 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)(3,7)(4,8)(5,14)(6,10)(9,15)(11,12)(13,16)(17,18,22,25)(19,26,23,20)(21,24,30,32)(27,31,29,28)(33,34,39,44)(35,42,37,40)(36,41,47,45)(38,46,48,43);(1,3)(2,7)(4,5)(6,11)(8,14)(9,16)(10,12)(13,15)(17,19,22,23)(18,20,25,26)(21,29,30,27)(24,31,32,28)(35,37)(38,48)(40,42)(43,46);(3,11)(4,13)(7,12)(8,16)(17,18,22,25)(19,20,23,26)(21,24,30,32)(27,28,29,31)(33,35,39,37)(34,40,44,42)(36,46,47,43)(38,45,48,41);(1,4,6,13)(2,8,10,16)(3,5,11,15)(7,14,12,9)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,36)(34,41)(35,43)(37,46)(38,42)(39,47)(40,48)(44,45);(1,5)(2,9)(3,4)(6,15)(7,16)(8,12)(10,14)(11,13)(17,20)(18,23)(19,25)(21,28)(22,26)(24,29)(27,32)(30,31)(33,37,39,35)(34,42,44,40)(36,43,47,46)(38,41,48,45);(1,2)(3,12)(4,8)(5,9)(6,10)(7,11)(13,16)(14,15)(17,21)(18,24)(19,27)(20,28)(22,30)(23,29)(25,32)(26,31)(33,38,39,48)(34,43,44,46)(35,45,37,41)(36,42,47,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8746330);