#8209993

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	#40790	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4104 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)(3,10)(4,14)(5,9)(6,7)(8,11)(12,13)(15,16)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(33,34)(35,42)(36,41)(37,38)(39,43)(40,46)(44,45)(47,48);(2,7)(3,11)(4,9)(13,16)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,48)(42,43);(2,7)(5,14)(8,10)(13,16)(19,25)(23,26)(27,29)(31,32)(33,36,39,38)(34,41,43,37)(35,44,46,47)(40,48,42,45);(1,3)(2,8)(4,12)(5,13)(6,11)(7,10)(9,15)(14,16)(17,20,22,18)(19,26,25,23)(21,28,30,24)(27,32,29,31)(33,37)(34,36)(35,45)(38,43)(39,41)(40,44)(42,47)(46,48);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,36)(34,37,43,41)(35,44,46,47)(40,48,42,45);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,35)(34,42)(36,47)(37,45)(38,44)(39,46)(40,43)(41,48)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=i,e^2=g,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, 8117189);