#8186909

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	#34967	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4083 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)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(34,36)(35,41)(40,44)(43,47);(1,3,8,6)(2,9,7,12)(4,14,5,13)(10,16,11,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,4,8,5)(2,10,7,11)(3,13,6,14)(9,15,12,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(1,5,8,4)(2,11,7,10)(3,14,6,13)(9,16,12,15)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,6,8,3)(2,12,7,9)(4,14,5,13)(10,16,11,15)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,7,8,2)(3,12,6,9)(4,11,5,10)(13,16,14,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

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

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, 8094105);