#8107587

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	#52561	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014783);