#8436772

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	#57490	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343968);