#8256618

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

3848 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:

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

Polycyclic presentation:

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

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, 8163814);