#10521322

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4113 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)(3,8,10,4)(5,9,14,15)(11,13,16,12)(17,18,24,26)(19,27,22,23)(20,29,21,25)(28,31,30,32)(33,34,38,42)(35,43,45,39)(36,46,44,40)(37,41,47,48);(2,7)(3,10)(9,15)(11,16)(17,19,24,22)(18,23,26,27)(20,28,21,30)(25,31,29,32)(33,35)(34,39)(36,47)(37,44)(38,45)(40,48)(41,46)(42,43);(17,20,24,21)(18,25,26,29)(19,28,22,30)(23,31,27,32)(33,36,38,44)(34,40,42,46)(35,37,45,47)(39,41,43,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,13)(12,15)(14,16)(17,21,24,20)(18,25,26,29)(19,28,22,30)(23,32,27,31)(33,35)(34,39)(36,37)(38,45)(40,41)(42,43)(44,47)(46,48);(1,4,6,8)(2,3,7,10)(5,13,14,12)(9,16,15,11)(17,22,24,19)(18,23,26,27)(20,30,21,28)(25,31,29,32)(33,37)(34,41)(35,44)(36,45)(38,47)(39,46)(40,43)(42,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,23)(18,22)(19,26)(20,31)(21,32)(24,27)(25,30)(28,29)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=hi,e^2=gi,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, 10428518);