#8416752

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

4153 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(19,25)(23,26)(27,29)(31,32)(33,34,38,39)(35,42,36,40)(37,41,45,46)(43,48,44,47);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18)(19,26)(20,28)(21,22)(23,25)(24,30)(27,31)(29,32)(34,39)(40,42)(41,46)(47,48);(4,12)(7,14)(10,16)(13,15)(18,22)(23,26)(24,28)(31,32)(33,35,38,36)(34,40,39,42)(37,44,45,43)(41,48,46,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,25)(18,23,22,26)(20,29,30,27)(24,32,28,31)(33,36,38,35)(34,40,39,42)(37,44,45,43)(41,47,46,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20,21,30)(18,24,22,28)(19,27,25,29)(23,31,26,32)(33,34)(35,40)(36,42)(37,41)(38,39)(43,47)(44,48)(45,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,37,38,45)(34,41,39,46)(35,43,36,44)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8323948);