#8093204

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	#2424	C₂²
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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8000400);