#8746795

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	#45408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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)(3,7,10,8)(4,9,12,13)(11,14,16,15)(18,22)(19,25)(24,27)(26,28)(33,34)(35,43)(36,42)(37,41)(38,44)(39,40)(45,48)(46,47);(1,3)(2,7)(4,11)(5,8)(6,10)(9,14)(12,16)(13,15)(18,22)(23,30)(24,27)(31,32)(34,40)(36,46)(41,43)(44,45);(3,10)(7,8)(11,16)(14,15)(17,18,21,22)(19,23,25,30)(20,24,29,27)(26,31,28,32)(33,35,39,37)(34,41,40,43)(36,44,46,45)(38,42,48,47);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,16)(13,14)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,36,39,46)(34,42,40,47)(35,45,37,44)(38,43,48,41);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,37,39,35)(34,43,40,41)(36,44,46,45)(38,42,48,47);(1,5,6,2)(3,8,10,7)(4,9,12,13)(11,14,16,15)(17,20,21,29)(18,24,22,27)(19,26,25,28)(23,31,30,32)(33,38,39,48)(34,44,40,45)(35,42,37,47)(36,41,46,43)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653991);