#8734802

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

4338 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)(3,5)(4,8)(6,9)(7,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34,39,41)(35,40,44,48)(36,37,46,42)(38,43,47,45);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,14)(13,16)(17,19,22,27)(18,23,26,32)(20,21,28,31)(24,30,29,25)(33,35)(34,40)(36,47)(37,45)(38,46)(39,44)(41,48)(42,43);(4,12)(8,16)(10,14)(13,15)(19,27)(20,28)(23,32)(24,29)(34,41)(35,44)(36,46)(43,45);(1,4,6,12)(2,8,9,16)(3,10,11,14)(5,13,7,15)(17,20)(18,24)(19,21)(22,28)(23,30)(25,32)(26,29)(27,31)(33,36)(34,42)(35,38)(37,41)(39,46)(40,45)(43,48)(44,47);(1,5,6,7)(2,3,9,11)(4,13,12,15)(8,10,16,14)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,32)(28,29)(33,37,39,42)(34,36,41,46)(35,45,44,43)(38,40,47,48);(1,3)(2,5)(4,14)(6,11)(7,9)(8,15)(10,12)(13,16)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,38)(34,43)(35,46)(36,44)(37,48)(39,47)(40,42)(41,45)

Polycyclic presentation:

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

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, 8641998);