#8291651

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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

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, 8198847);