#10481003

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=i,f^2=g,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, 10388199);