#8944050

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(18,19)(23,25)(24,26)(30,31)(33,34,38,41)(35,39,45,48)(36,40,46,37)(42,43,44,47);(2,6)(7,9)(8,12)(14,15)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,35)(34,39)(36,44)(37,43)(38,45)(40,47)(41,48)(42,46);(2,6)(4,13)(7,9)(10,16)(20,27)(21,29)(23,25)(24,26)(33,36,38,46)(34,40,41,37)(35,42,45,44)(39,43,48,47);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,37)(34,36)(35,43)(38,40)(39,44)(41,46)(42,48)(45,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36)(34,40)(35,44)(37,41)(38,46)(39,47)(42,45)(43,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,35)(34,39)(36,44)(37,43)(38,45)(40,47)(41,48)(42,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8851246);