#8741957

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

3945 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8649153);