#10247139

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=hi,e^2=h,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, 10154335);