#8926706

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3855 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)(3,9)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(17,18)(19,23)(20,30)(21,25)(22,26)(24,28)(27,29)(31,32)(34,39)(37,45)(40,46)(44,48);(2,7)(3,10)(8,13)(11,16)(19,27)(21,31)(23,29)(25,32)(36,43)(40,46)(42,47)(44,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,15)(14,16)(17,19,22,27)(18,23,26,29)(20,21,28,31)(24,32,30,25)(33,34,38,39)(35,37,41,45)(36,40,43,46)(42,44,47,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,20)(18,24)(19,21)(22,28)(23,32)(25,29)(26,30)(27,31)(33,35,38,41)(34,37,39,45)(36,42,43,47)(40,44,46,48);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,32)(29,30)(33,36,38,43)(34,40,39,46)(35,42,41,47)(37,44,45,48);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,18)(19,29)(20,24)(21,25)(22,26)(23,27)(28,30)(31,32)(33,37)(34,41)(35,39)(36,44)(38,45)(40,47)(42,46)(43,48)

Polycyclic presentation:

2^9|c^2=hi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833902);