#8739446

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3814 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)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,34,39,42)(35,37,41,38)(36,44,47,40)(43,46,48,45);(1,3,6,11)(2,7,9,4)(5,10,15,13)(8,14,16,12)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35,39,41)(34,38,42,37)(36,43,47,48)(40,46,44,45);(1,3,6,11)(2,7,9,4)(5,10,15,13)(8,14,16,12)(17,19,23,21)(18,25,20,24)(22,27,30,29)(26,32,28,31)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,46)(47,48);(1,3)(2,4)(5,13)(6,11)(7,9)(8,14)(10,15)(12,16)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,36,39,47)(34,40,42,44)(35,43,41,48)(37,45,38,46);(1,4)(2,3)(5,14)(6,7)(8,13)(9,11)(10,16)(12,15)(17,21,23,19)(18,24,20,25)(22,29,30,27)(26,31,28,32)(33,37,39,38)(34,41,42,35)(36,45,47,46)(40,48,44,43);(1,5)(2,8)(3,10)(4,12)(6,15)(7,14)(9,16)(11,13)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,38,39,37)(34,35,42,41)(36,46,47,45)(40,43,44,48)

Polycyclic presentation:

2^9|a^2=hi,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, 8646642);