#8348936

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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:

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

Polycyclic presentation:

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

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, 8256132);