#8425664

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	#52541	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8332860);