#8159499

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	#43275	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(21,29)(24,30)(26,31)(27,32)(34,38)(36,44)(40,42)(43,45);(2,7)(3,9)(10,13)(11,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32)(33,34,39,38)(35,40,37,42)(36,41,44,46)(43,47,45,48);(2,7)(3,9)(10,13)(11,16)(18,23)(19,25)(24,30)(26,31)(33,35,39,37)(34,40,38,42)(36,45,44,43)(41,48,46,47);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,37,39,35)(34,42,38,40)(36,43,44,45)(41,47,46,48);(1,5,6,14)(2,10,7,13)(3,11,9,16)(4,12,8,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,38,39,34)(35,42,37,40)(36,46,44,41)(43,48,45,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=i,e^2=g,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, 8066695);