#8930252

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

4053 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=ghi,e^2=h,f^2=g,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, 8837448);