#8745909

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	#52540	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3951 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653105);