#10360960

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	#38334	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=ghi,e^2=gh,f^2=gh,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, 10268156);