#10360884

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

3951 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10268080);