#8933123

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	#46640	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 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)(4,6)(5,8)(9,11)(10,14)(12,13)(15,16)(17,18,22,23)(19,24,25,20)(21,29,30,26)(27,28,32,31)(35,41)(36,44)(39,42)(40,45);(2,4)(3,9)(8,12)(10,15)(18,23)(19,25)(26,29)(27,32)(33,34)(35,42)(36,45)(37,38)(39,41)(40,44)(43,47)(46,48);(3,9)(5,13)(7,11)(8,12)(18,23)(19,25)(21,30)(28,31)(34,38)(35,41)(40,45)(43,46);(1,3,6,9)(2,7,4,11)(5,10,13,15)(8,14,12,16)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,29)(30,32)(33,35)(34,39)(36,46)(37,41)(38,42)(40,48)(43,44)(45,47);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(33,36)(34,40)(35,43)(37,44)(38,45)(39,47)(41,46)(42,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48)

Polycyclic presentation:

2^9|a^2=h,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, 8840319);