#8630904

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=g,e^2=hi,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, 8538100);