#8979772

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

4221 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8886968);