#8928723

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

4139 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,26)(19,24,27,32)(20,30,28,22)(21,25,31,29)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,44)(47,48);(2,6)(7,10)(8,13)(14,16)(17,19)(18,24)(20,21)(22,29)(23,27)(25,30)(26,32)(28,31)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(3,9)(4,12)(7,10)(8,13)(19,27)(20,28)(22,30)(24,32)(33,34,38,41)(35,36,39,42)(37,45,46,40)(43,48,47,44);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,20)(18,22)(19,21)(23,28)(24,29)(25,32)(26,30)(27,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,29)(30,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,22)(18,20)(19,29)(21,24)(23,30)(25,27)(26,28)(31,32)(33,37,38,46)(34,40,41,45)(35,43,39,47)(36,44,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8835919);