#8932770

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	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,26)(20,22)(21,24)(23,25)(27,32)(28,29)(30,31)(35,41)(36,44)(39,42)(40,45);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,19,23,21)(18,24,25,26)(20,30,27,28)(22,29,32,31)(33,34)(35,42)(36,45)(37,38)(39,41)(40,44)(43,47)(46,48);(4,12)(7,14)(10,16)(13,15)(18,25)(22,32)(24,26)(29,31)(34,38)(39,42)(40,45)(47,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20,23,27)(18,22,25,32)(19,28,21,30)(24,31,26,29)(33,35)(34,39)(36,46)(37,41)(38,42)(40,48)(43,44)(45,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,21,23,19)(18,26,25,24)(20,28,27,30)(22,31,32,29)(33,36)(34,40)(35,43)(37,44)(38,45)(39,47)(41,46)(42,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,22)(18,27)(19,29)(20,25)(21,31)(23,32)(24,28)(26,30)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839966);