#8463650

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	#40359	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8370846);