#8107678

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	#43298	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014874);