#10407790

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	#46513	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=i,e^2=hi,f^2=hi,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, 10314986);