#8360937

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	#48990	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,d^2=hi,e^2=gh,f^2=h,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, 8268133);