#8479496

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	#36795	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386692);