#8633840

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	#44129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,e^2=h,f^2=hi,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, 8541036);