#8932821

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=h,f^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, 8840017);