#8417031

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	#2818	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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)(3,8,12,15)(4,13,14,9)(5,10,16,11)(17,18)(19,26)(20,23)(21,28)(22,25)(24,30)(27,29)(31,32)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(2,7)(4,14)(8,15)(10,11)(18,20)(19,25)(24,28)(27,31)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(2,7)(5,16)(8,15)(9,13)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,3)(2,8)(4,10)(5,9)(6,12)(7,15)(11,14)(13,16)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,4)(2,9)(3,11)(5,15)(6,14)(7,13)(8,16)(10,12)(17,21,23,30)(18,24,20,28)(19,27,25,31)(22,29,26,32)(33,34,39,38)(35,40,37,42)(36,41,46,45)(43,47,44,48);(1,5,6,16)(2,10,7,11)(3,9,12,13)(4,15,14,8)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8324227);