#10407804

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	#47410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4046 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=ghi,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, 10315000);