#8739397

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3831 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 8646593);