#8324178

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	#56544	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8231374);