#8630900

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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:

(1,2,7,9)(3,5,6,8)(4,12,15,10)(11,16,13,14)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,38,41)(35,37,40,47)(36,43,46,39)(42,48,44,45);(1,3,7,6)(2,8,9,5)(4,13,15,11)(10,14,12,16)(18,19)(23,25)(24,26)(30,31)(33,35,38,40)(34,37,41,47)(36,44,46,42)(39,48,43,45);(2,9)(5,8)(10,12)(14,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,36,38,46)(34,39,41,43)(35,42,40,44)(37,45,47,48);(1,4,7,15)(2,10,9,12)(3,11,6,13)(5,14,8,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35)(34,37)(36,42)(38,40)(39,45)(41,47)(43,48)(44,46);(1,5,7,8)(2,3,9,6)(4,14,15,16)(10,11,12,13)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,43)(47,48);(1,6,7,3)(2,8,9,5)(4,11,15,13)(10,14,12,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(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=i,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, 8538096);