#8739399

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3824 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,d^2=gh,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, 8646595);