#10246915

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154111);