#8746754

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8653950);