#10352523

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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)(3,11,12,8)(4,9,10,5)(13,14,16,15)(17,18,22,20)(19,23,25,27)(21,24,30,28)(26,31,29,32)(33,34)(35,38)(36,41)(37,42)(39,43)(40,45)(44,46)(47,48);(2,7)(3,12)(5,9)(13,16)(17,19)(18,23)(20,27)(21,29)(22,25)(24,32)(26,30)(28,31)(33,35,39,45)(34,40,43,38)(36,44,47,37)(41,42,48,46);(2,7)(4,10)(8,11)(13,16)(19,25)(23,27)(26,29)(31,32)(33,36,39,47)(34,41,43,48)(35,44,45,37)(38,46,40,42);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,15)(10,16)(17,20,22,18)(19,23,25,27)(21,28,30,24)(26,31,29,32)(33,35)(34,40)(36,44)(37,47)(38,43)(39,45)(41,42)(46,48);(1,4,6,10)(2,9,7,5)(3,13,12,16)(8,15,11,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,37,39,44)(34,42,43,46)(35,36,45,47)(38,41,40,48);(1,5,6,9)(2,10,7,4)(3,14,12,15)(8,16,11,13)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,38,39,40)(34,35,43,45)(36,46,47,42)(37,41,44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10259719);