#8622060

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	#2657	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529256);