#10246877

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	#45593	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34,38,40)(35,36,43,39)(37,45,47,41)(42,48,46,44);(2,7)(3,10)(9,16)(11,12)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,43)(34,39,40,36)(37,42,47,46)(41,48,45,44);(18,22)(23,25)(24,28)(30,31)(34,40)(36,39)(41,45)(44,48);(1,3,6,10)(2,8,7,5)(4,12,14,11)(9,15,16,13)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,38,39)(34,35,40,43)(37,44,47,48)(41,42,45,46);(1,4,6,14)(2,9,7,16)(3,11,10,12)(5,15,8,13)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,35)(34,36)(37,46)(38,43)(39,40)(41,48)(42,47)(44,45);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,11,16,12)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,37,38,47)(34,41,40,45)(35,42,43,46)(36,44,39,48)

Polycyclic presentation:

2^9|a^2=g,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, 10154073);