#10504492

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	#51942	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,16)(14,15)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34)(35,39)(36,46)(37,41)(38,42)(40,47)(43,45)(44,48);(2,7)(3,9)(10,15)(11,16)(17,18,23,20)(19,24,22,26)(21,25,30,28)(27,31,29,32)(33,35)(34,39)(36,37)(38,45)(40,48)(41,46)(42,43)(44,47);(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,45)(44,48)(46,47);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,13,15,12)(17,20,23,18)(19,26,22,24)(21,25,30,28)(27,31,29,32)(33,36)(34,40)(35,44)(37,45)(38,47)(39,41)(42,46)(43,48);(1,4,6,8)(2,9,7,3)(5,13,14,12)(10,11,15,16)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,37,38,44)(34,41,42,48)(35,36,45,47)(39,46,43,40);(1,5,6,14)(2,10,7,15)(3,11,9,16)(4,12,8,13)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,34)(35,39)(36,40)(37,48)(38,42)(41,44)(43,45)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,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, 10411688);