#8656222

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	#45649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4088 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8563418);