#8193822

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	#57185	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4079 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)(3,5)(4,12)(6,8)(7,11)(9,14)(10,15)(13,16)(33,34)(35,36)(37,45)(38,41)(39,44)(40,46)(42,48)(43,47);(1,3)(2,7)(4,10)(5,8)(6,11)(9,15)(12,13)(14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,39)(34,36,41,44)(37,42,46,47)(40,43,45,48);(2,8)(5,7)(9,12)(13,15)(18,22)(23,25)(24,29)(30,31)(33,36,38,44)(34,39,41,35)(37,43,46,48)(40,47,45,42);(1,3)(2,7)(4,10)(5,8)(6,11)(9,15)(12,13)(14,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,4,6,14)(2,9,8,12)(3,10,11,16)(5,13,7,15)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37)(34,40)(35,42)(36,43)(38,46)(39,47)(41,45)(44,48);(1,5,6,7)(2,3,8,11)(4,13,14,15)(9,10,12,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34)(35,36)(37,40)(38,41)(39,44)(42,43)(45,46)(47,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=ghi,e^2=hi,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, 8101018);