#8429437

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	#53044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4103 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8336633);