#8889055

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	#2819	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(3,5)(4,12)(6,7)(8,11)(9,15)(10,16)(13,14)(19,25)(22,26)(27,31)(30,32)(33,34,39,36)(35,41,37,40)(38,44,46,42)(43,47,45,48);(2,7)(5,8)(9,12)(14,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,32)(30,31)(33,34,39,36)(35,41,37,40)(38,44,46,42)(43,47,45,48);(1,3,6,11)(2,5,7,8)(4,10,15,13)(9,14,12,16)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,3)(2,8)(4,13)(5,7)(6,11)(9,14)(10,15)(12,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,29)(30,31)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,4)(2,9)(3,10)(5,14)(6,15)(7,12)(8,16)(11,13)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,5,6,8)(2,3,7,11)(4,14,15,16)(9,10,12,13)(17,20,21,29)(18,23,24,28)(19,27,25,31)(22,30,26,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=h,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8796251);