#8602874

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	#39728	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4154 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8510070);