#9039125

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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)(3,8)(4,12)(5,7)(6,11)(9,13)(10,14)(15,16)(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(33,34,38,39)(35,42,37,40)(36,41,44,46)(43,48,45,47);(1,3)(2,6)(4,10)(5,11)(7,8)(9,14)(12,15)(13,16)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(34,39)(36,44)(40,42)(43,45);(2,7)(4,13)(6,8)(10,16)(17,18,22,20)(19,23,27,24)(21,28,29,25)(26,31,32,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,3)(2,8)(4,10)(5,11)(6,7)(9,15)(12,14)(13,16)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,35,38,37)(34,40,39,42)(36,43,44,45)(41,47,46,48);(1,2)(3,6)(4,9)(5,7)(8,11)(10,14)(12,13)(15,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36)(34,41)(35,43)(37,45)(38,44)(39,46)(40,47)(42,48);(1,4,5,13)(2,9,7,12)(3,10,11,16)(6,14,8,15)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,37,38,35)(34,42,39,40)(36,45,44,43)(41,48,46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=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, 8946321);