#8393368

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	#56473	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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)(3,7)(4,9)(5,13)(6,8)(10,16)(11,12)(14,15)(33,34,40,43)(35,44,39,41)(36,38,37,42)(45,47,46,48);(1,3,6,9)(2,7,8,4)(5,14,16,11)(10,12,13,15)(17,18,22,23)(19,25,28,24)(20,29,31,21)(26,32,30,27)(33,35,40,39)(34,41,43,44)(36,46,37,45)(38,47,42,48);(2,8)(3,9)(10,13)(11,14)(18,23)(21,29)(24,25)(27,32)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,47,44,48);(1,4,6,7)(2,9,8,3)(5,15,16,12)(10,11,13,14)(17,19,22,28)(18,24,23,25)(20,30,31,26)(21,27,29,32)(33,37,40,36)(34,38,43,42)(35,46,39,45)(41,48,44,47);(1,5,6,16)(2,10,8,13)(3,11,9,14)(4,12,7,15)(17,20)(18,21)(19,26)(22,31)(23,29)(24,32)(25,27)(28,30)(33,38)(34,36)(35,47)(37,43)(39,48)(40,42)(41,46)(44,45);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,14)(13,16)(17,21)(18,20)(19,27)(22,29)(23,31)(24,30)(25,26)(28,32)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,48,42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300564);