#8734505

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	#52748	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=h,e^2=g,f^2=g,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, 8641701);