#8228358

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

4100 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:

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

Polycyclic presentation:

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

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, 8135554);