#10069010

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4299 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=i,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9976206);