#8107610

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	#40679	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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)(18,24)(20,28)(22,26)(27,29)(33,34,38,42)(35,41,37,39)(36,44,46,40)(43,47,45,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,18)(19,22)(20,25)(21,26)(23,24)(27,30)(28,31)(29,32)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,48)(45,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,23,21)(18,22,24,26)(20,29,28,27)(25,32,31,30)(33,35,38,37)(34,39,42,41)(36,45,46,43)(40,48,44,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20,23,28)(18,25,24,31)(19,27,21,29)(22,30,26,32)(33,36,38,46)(34,40,42,44)(35,43,37,45)(39,47,41,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,21,23,19)(18,26,24,22)(20,29,28,27)(25,32,31,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,46)(45,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,22,23,26)(18,19,24,21)(20,30,28,32)(25,27,31,29)(33,37,38,35)(34,41,42,39)(36,45,46,43)(40,48,44,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,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, 8014806);