#8746520

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	#42759	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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)(3,7)(4,9)(5,10)(6,11)(8,12)(13,15)(14,16)(17,18)(19,29)(20,26)(21,22)(23,28)(24,31)(25,32)(27,30)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,48)(46,47);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(17,19)(18,24)(20,21)(22,25)(23,31)(26,27)(28,29)(30,32)(33,35,38,44)(34,39,42,43)(36,41,45,46)(37,40,47,48);(3,12)(7,8)(13,16)(14,15)(17,20,23,32)(18,25,28,26)(19,21,31,30)(22,29,27,24)(33,36,38,45)(34,40,42,48)(35,41,44,46)(37,39,47,43);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,20)(18,26)(19,30)(21,31)(22,24)(23,32)(25,28)(27,29)(33,36)(34,40)(35,41)(37,43)(38,45)(39,47)(42,48)(44,46);(1,4)(2,9)(3,13)(5,11)(6,10)(7,15)(8,14)(12,16)(17,21,23,30)(18,27,28,22)(19,20,31,32)(24,26,29,25)(33,34)(35,39)(36,40)(37,46)(38,42)(41,47)(43,44)(45,48);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,22,23,27)(18,21,28,30)(19,25,31,26)(20,29,32,24)(33,37,38,47)(34,41,42,46)(35,40,44,48)(36,39,45,43)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653716);