#10504583

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	#40413	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=g,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, 10411779);