#8712475

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	#54921	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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)(3,7)(4,13)(5,10)(6,8)(9,14)(11,15)(12,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(34,38)(35,41)(40,43)(42,44);(1,3,6,11)(2,7,8,15)(4,12,14,10)(5,13,16,9)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,34,37,38)(35,39,41,46)(36,40,45,43)(42,47,44,48);(2,8)(3,11)(9,13)(10,12)(19,28)(20,29)(24,30)(25,32)(33,34,37,38)(35,39,41,46)(36,40,45,43)(42,47,44,48);(1,3)(2,7)(4,12)(5,9)(6,11)(8,15)(10,14)(13,16)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(1,4,6,14)(2,9,8,13)(3,12,11,10)(5,15,16,7)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,5)(2,10)(3,13)(4,7)(6,16)(8,12)(9,11)(14,15)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,36,37,45)(34,40,38,43)(35,42,41,44)(39,47,46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8619671);