#8271164

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

3926 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=hi,e^2=hi,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, 8178360);