#10455675

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

4155 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)(3,8,10,4)(5,9,14,15)(11,13,16,12)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,29)(28,32)(34,40)(35,36)(38,42)(44,46);(2,7)(3,10)(9,15)(11,16)(17,19,23,24)(18,21,26,20)(22,27,31,32)(25,29,30,28)(33,34,39,40)(35,43,36,41)(37,42,48,38)(44,45,46,47);(17,20,23,21)(18,24,26,19)(22,28,31,29)(25,32,30,27)(33,35,39,36)(34,41,40,43)(37,44,48,46)(38,45,42,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,13)(12,15)(14,16)(17,21,23,20)(18,24,26,19)(22,29,31,28)(25,32,30,27)(33,36,39,35)(34,41,40,43)(37,44,48,46)(38,47,42,45);(1,4,6,8)(2,3,7,10)(5,13,14,12)(9,16,15,11)(17,18)(19,21)(20,24)(22,30)(23,26)(25,31)(27,28)(29,32)(33,37)(34,42)(35,44)(36,46)(38,40)(39,48)(41,47)(43,45);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,22,23,31)(18,25,26,30)(19,27,24,32)(20,28,21,29)(33,38,39,42)(34,37,40,48)(35,45,36,47)(41,44,43,46)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10362871);