#8762065

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	#47995	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4197 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8669261);