#8159191

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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)(3,4,10,8)(5,13,14,9)(11,15,16,12)(17,18,22,24)(19,23,25,20)(21,29,30,26)(27,28,32,31)(33,34,39,38)(35,42,37,40)(36,41,44,46)(43,48,45,47);(2,7)(3,10)(9,13)(11,16)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,31,29,28)(36,44)(41,46)(43,45)(47,48);(2,7)(3,10)(9,13)(11,16)(18,24)(19,25)(26,29)(27,32)(33,35,39,37)(34,40,38,42)(36,43,44,45)(41,47,46,48);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,19)(18,23)(20,24)(21,27)(22,25)(26,31)(28,29)(30,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20,22,23)(18,25,24,19)(21,28,30,31)(26,32,29,27)(33,37,39,35)(34,42,38,40)(36,45,44,43)(41,48,46,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21,22,30)(18,26,24,29)(19,27,25,32)(20,28,23,31)(33,38,39,34)(35,42,37,40)(36,46,44,41)(43,48,45,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066387);