#8889117

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

4171 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)(3,9)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(17,18,22,23)(19,26,24,20)(21,29,30,25)(27,28,31,32)(35,41)(38,42)(43,47)(46,48);(2,6)(7,9)(8,12)(15,16)(18,23)(20,26)(21,30)(27,31)(33,34)(35,38)(36,44)(37,40)(39,45)(41,42)(43,48)(46,47);(3,10)(7,9)(11,13)(15,16)(17,19,22,24)(18,20,23,26)(21,27,30,31)(25,28,29,32)(33,34,37,40)(35,38,41,42)(36,39,45,44)(43,46,47,48);(1,3)(2,7)(4,13)(5,10)(6,9)(8,16)(11,14)(12,15)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,45)(46,47);(1,4,5,14)(2,8,6,12)(3,11,10,13)(7,15,9,16)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,29)(30,32)(33,35,37,41)(34,38,40,42)(36,43,45,47)(39,46,44,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,21,22,30)(18,25,23,29)(19,27,24,31)(20,28,26,32)(33,36,37,45)(34,39,40,44)(35,43,41,47)(38,46,42,48)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,e^2=ghi,f^2=hi,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, 8796313);