#8932123

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	#55589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839319);