#8188180

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	#42591	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 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)(17,18)(19,23)(20,25)(21,29)(22,24)(26,30)(27,28)(31,32)(34,37)(35,43)(36,46)(42,48);(1,3)(2,7)(4,10)(5,8)(6,11)(9,15)(12,13)(14,16)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,31,29,28)(33,34,39,37)(35,44,43,40)(36,47,46,41)(38,42,45,48);(2,8)(5,7)(9,12)(13,15)(18,24)(19,25)(26,29)(27,32)(33,35)(34,40)(36,45)(37,44)(38,46)(39,43)(41,48)(42,47);(1,3)(2,7)(4,10)(5,8)(6,11)(9,15)(12,13)(14,16)(17,20,22,23)(18,25,24,19)(21,28,30,31)(26,32,29,27)(33,36,39,46)(34,41,37,47)(35,45,43,38)(40,48,44,42);(1,4,6,14)(2,9,8,12)(3,10,11,16)(5,13,7,15)(17,21,22,30)(18,26,24,29)(19,27,25,32)(20,28,23,31)(33,37,39,34)(35,44,43,40)(36,47,46,41)(38,48,45,42);(1,5,6,7)(2,3,8,11)(4,13,14,15)(9,10,12,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,32)(29,30)(33,38,39,45)(34,42,37,48)(35,46,43,36)(40,47,44,41)

Polycyclic presentation:

2^9|b^2=i,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, 8095376);