#8271079

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

3847 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=i,d^2=hi,e^2=ghi,f^2=gi,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, 8178275);