#8691779

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4159 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8598975);