#10216621

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	#40092	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,23)(19,27,28,24)(20,25,29,32)(21,30,31,26)(33,34)(35,40)(36,37)(38,46)(39,41)(42,48)(43,45)(44,47);(3,9)(7,13)(10,15)(14,16)(18,23)(20,29)(24,27)(26,30)(33,35,39,36)(34,40,41,37)(38,43,48,44)(42,47,46,45);(2,6)(7,13)(8,11)(14,16)(18,23)(19,28)(25,32)(26,30)(34,41)(35,36)(42,46)(43,44);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,24)(20,30)(21,25)(22,28)(23,27)(26,29)(31,32)(33,36,39,35)(34,37,41,40)(38,44,48,43)(42,45,46,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,25)(19,26)(21,27)(22,29)(23,32)(24,31)(28,30)(33,37)(34,36)(35,41)(38,47)(39,40)(42,43)(44,46)(45,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,31)(18,26,23,30)(19,25,28,32)(20,24,29,27)(33,38,39,48)(34,42,41,46)(35,43,36,44)(37,45,40,47)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10123817);