#8630823

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	#54019	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8538019);