#8968123

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=hi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8875319);