#10340222

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	#55471	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,e^2=gh,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, 10247418);