#8630920

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	#48373	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4211 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538116);