#8757450

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	#50368	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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)(3,10,5,8)(4,12,14,9)(11,15,13,16)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34)(35,41)(36,44)(37,42)(38,39)(40,46)(43,45)(47,48);(2,7)(8,10)(9,12)(15,16)(21,29)(25,28)(27,30)(31,32)(34,39)(35,42)(40,44)(43,48);(1,3,6,5)(2,8,7,10)(4,13,14,11)(9,16,12,15)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,47)(46,48);(1,4,6,14)(2,9,7,12)(3,11,5,13)(8,15,10,16)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,36,38,46)(34,40,39,44)(35,43,42,48)(37,45,41,47);(1,5,6,3)(2,10,7,8)(4,13,14,11)(9,16,12,15)(17,21,23,29)(18,25,20,28)(19,27,22,30)(24,31,26,32)(33,34)(35,37)(36,40)(38,39)(41,42)(43,45)(44,46)(47,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(17,22,23,19)(18,26,20,24)(21,30,29,27)(25,32,28,31)(33,37)(34,35)(36,45)(38,41)(39,42)(40,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8664646);