#8696671

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	#59992	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4130 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=g,e^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, 8603867);