#8633357

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	#45599	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4186 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=gh,e^2=gh,f^2=gi,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, 8540553);