#8933069

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	#59576	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4082 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:

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

Polycyclic presentation:

2^9|a^2=h,c^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,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, 8840265);