#10504493

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	#50979	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3910 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gh,e^2=ghi,f^2=gh,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, 10411689);