#8159822

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	#58027	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8067018);