#8534790

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	#55328	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4136 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8441986);