#8712329

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	#40392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4101 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)(3,11,12,8)(4,13,14,9)(5,10,15,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34)(35,44)(36,37)(38,42)(39,43)(40,47)(41,46)(45,48);(2,7)(3,12)(9,13)(10,16)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,35)(34,40)(36,45)(37,42)(38,46)(39,47)(41,48)(43,44);(2,7)(3,12)(9,13)(10,16)(19,28)(20,29)(24,30)(25,32)(33,36,39,46)(34,37,43,41)(35,45,47,38)(40,42,44,48);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,9,15,13)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,36)(34,41)(35,45)(37,43)(38,47)(39,46)(40,48)(42,44);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,8,15,11)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,37,39,41)(34,36,43,46)(35,42,47,48)(38,40,45,44);(1,5,6,15)(2,10,7,16)(3,13,12,9)(4,11,14,8)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,38,39,45)(34,42,43,48)(35,46,47,36)(37,44,41,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=h,e^2=g,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, 8619525);