#10455714

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	#49741	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3908 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)(20,28)(23,29)(26,32)(30,31)(33,34)(35,42)(36,39)(37,45)(38,41)(40,47)(43,44)(46,48);(2,3)(7,9)(8,10)(15,16)(17,18)(19,24)(20,29)(21,25)(22,27)(23,28)(26,30)(31,32)(33,34,38,41)(35,36,39,42)(37,40,47,45)(43,44,48,46);(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,32)(29,31)(33,35,38,39)(34,36,41,42)(37,43,47,48)(40,44,45,46);(1,3,6,2)(4,9,12,7)(5,10,14,8)(11,16,13,15)(17,20)(18,23)(19,26)(21,28)(22,30)(24,31)(25,29)(27,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,32)(29,30)(33,36)(34,39)(35,41)(37,46)(38,42)(40,43)(44,47)(45,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,29)(31,32)(33,37)(34,40)(35,43)(36,44)(38,47)(39,48)(41,45)(42,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gh,e^2=gh,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, 10362910);