#8734945

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	#44437	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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)(3,11)(4,5)(6,8)(7,9)(10,15)(12,13)(14,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34,39,43)(35,41,37,40)(36,42,47,38)(44,45,46,48);(1,3,7,6)(2,8,9,11)(4,14,10,12)(5,13,15,16)(17,19,22,27)(18,23,26,32)(20,21,28,31)(24,30,29,25)(33,35,39,37)(34,40,43,41)(36,44,47,46)(38,45,42,48);(2,9)(4,10)(8,11)(12,14)(19,27)(20,28)(23,32)(24,29)(33,35,39,37)(34,41,43,40)(36,46,47,44)(38,45,42,48);(1,4,7,10)(2,5,9,15)(3,12,6,14)(8,16,11,13)(17,20)(18,24)(19,21)(22,28)(23,30)(25,32)(26,29)(27,31)(33,36,39,47)(34,42,43,38)(35,44,37,46)(40,48,41,45);(1,5)(2,10)(3,13)(4,9)(6,16)(7,15)(8,12)(11,14)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,32)(28,29)(33,37,39,35)(34,40,43,41)(36,46,47,44)(38,45,42,48);(1,6,7,3)(2,8,9,11)(4,12,10,14)(5,13,15,16)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,38,39,42)(34,36,43,47)(35,45,37,48)(40,44,41,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,e^2=gh,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, 8642141);