#10106632

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	#53534	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=gh,e^2=g,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, 10013828);