#8159261

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3885 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066457);