#8930356

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

4113 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gi,e^2=g,f^2=h,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, 8837552);