#8345915

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	#53063	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=hi,e^2=hi,f^2=gi,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, 8253111);