#8093488

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	#2815	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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)(4,9)(5,12)(6,7)(10,15)(11,14)(13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34)(35,43)(36,38)(37,42)(39,40)(41,47)(44,45)(46,48);(3,11)(4,13)(8,14)(9,16)(18,22)(23,25)(24,29)(30,31)(34,40)(36,42)(41,43)(44,48);(2,7)(3,11)(9,16)(10,12)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,39,47)(34,41,40,43)(36,44,42,48)(37,45,38,46);(1,3)(2,8)(4,12)(5,9)(6,11)(7,14)(10,13)(15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36)(34,37)(35,44)(38,40)(39,42)(41,45)(43,46)(47,48);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,14,15,8)(33,37,39,38)(34,42,40,36)(35,45,47,46)(41,48,43,44);(1,5)(2,10)(3,9)(4,14)(6,15)(7,12)(8,13)(11,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,38,39,37)(34,36,40,42)(35,46,47,45)(41,44,43,48)

Polycyclic presentation:

2^9|c^2=gi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

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, 8000684);