#10040844

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3733 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=ghi,e^2=h,f^2=h,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, 9948040);