#8406489

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	#52733	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=gh,e^2=gh,f^2=h,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, 8313685);