#8156968

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

4238 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)(3,8)(4,10)(5,14)(6,13)(7,9)(11,15)(12,16)(20,28)(23,30)(26,32)(29,31)(33,34,38,42)(35,43,45,39)(36,40,46,48)(37,47,44,41);(1,3)(2,8)(4,7)(5,13)(6,14)(9,10)(11,16)(12,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,35,38,45)(34,39,42,43)(36,44,46,37)(40,47,48,41);(2,9)(3,4)(11,14)(12,13)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,36)(34,40)(35,44)(37,45)(38,46)(39,47)(41,43)(42,48);(1,4,7,3)(2,10,9,8)(5,13,15,12)(6,11,16,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,37)(34,41)(35,36)(38,44)(39,40)(42,47)(43,48)(45,46);(1,5)(2,11)(3,13)(4,12)(6,10)(7,15)(8,16)(9,14)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36)(34,40)(35,44)(37,45)(38,46)(39,47)(41,43)(42,48);(1,6)(2,12)(3,14)(4,11)(5,10)(7,16)(8,15)(9,13)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8064164);