#8751272

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3763 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)(3,12)(4,10)(5,13)(6,8)(7,9)(11,15)(14,16)(18,22)(20,27)(23,28)(26,31)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,3,7,6)(2,8,9,12)(4,13,14,11)(5,10,15,16)(19,25)(23,28)(26,31)(30,32)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(2,9)(4,14)(8,12)(11,13)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,4)(2,10)(3,11)(5,8)(6,13)(7,14)(9,16)(12,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,5)(2,11)(3,10)(4,8)(6,16)(7,15)(9,13)(12,14)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47);(1,6,7,3)(2,8,9,12)(4,11,14,13)(5,10,15,16)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=i,e^2=gh,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, 8658468);