#8979752

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
prime radical	#50594	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=h,e^2=gh,f^2=h,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, 8886948);