#8695814

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	#51017	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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)(3,6)(4,10)(5,15)(7,9)(8,11)(12,13)(14,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(1,3,8,7)(2,9,11,6)(4,12,5,14)(10,16,15,13)(19,28)(20,29)(24,30)(25,32)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(1,4,8,5)(2,10,11,15)(3,12,7,14)(6,13,9,16)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,48,43,41);(1,5,8,4)(2,10,11,15)(3,12,7,14)(6,16,9,13)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,6)(2,3)(4,13)(5,16)(7,11)(8,9)(10,12)(14,15)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,7,8,3)(2,9,11,6)(4,14,5,12)(10,16,15,13)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603010);