#8933132

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	#46538	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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)(3,8)(4,10)(5,13)(6,7)(9,14)(11,12)(15,16)(17,18,22,23)(19,24,26,21)(20,28,29,25)(27,31,30,32)(35,41)(36,44)(39,42)(40,45);(2,7)(3,10)(9,13)(11,16)(18,23)(19,26)(20,29)(31,32)(33,34)(35,42)(36,45)(37,38)(39,41)(40,44)(43,47)(46,48);(2,7)(3,10)(5,14)(12,15)(19,26)(21,24)(27,30)(31,32)(34,38)(35,41)(40,45)(43,46);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,19)(18,24)(20,30)(21,23)(22,26)(25,31)(27,29)(28,32)(33,35)(34,39)(36,46)(37,41)(38,42)(40,48)(43,44)(45,47);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20,22,29)(18,25,23,28)(19,27,26,30)(21,31,24,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,47)(41,46)(42,48);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,21,22,24)(18,19,23,26)(20,31,29,32)(25,27,28,30)(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=h,e^2=g,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,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, 8840328);