#8517606

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	#37483	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4286 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^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, 8424802);