#8931045

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4152 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=gh,e^2=gi,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, 8838241);