#8933124

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

4145 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)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,18,22,23)(19,21,25,31)(20,27,28,24)(26,32,29,30)(35,41)(36,44)(39,42)(40,45);(2,6)(7,9)(8,12)(14,15)(18,23)(19,25)(20,28)(30,32)(33,34)(35,42)(36,45)(37,38)(39,41)(40,44)(43,47)(46,48);(2,6)(4,13)(7,9)(10,16)(19,25)(21,31)(26,29)(30,32)(34,38)(35,41)(40,45)(43,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,22,25)(18,21,23,31)(20,29,28,26)(24,32,27,30)(33,35)(34,39)(36,46)(37,41)(38,42)(40,48)(43,44)(45,47);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20,22,28)(18,24,23,27)(19,26,25,29)(21,30,31,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,47)(41,46)(42,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21)(18,25)(19,23)(20,30)(22,31)(24,29)(26,27)(28,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48)

Polycyclic presentation:

2^9|a^2=gh,d^2=gh,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,d^c=dgi,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, 8840320);