#10519364

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	#43298	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3906 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=hi,e^2=ghi,f^2=gh,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, 10426560);