#8406864

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	#41495	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18,23,29)(19,28,31,24)(20,27,22,25)(21,26,32,30)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,45)(47,48);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(17,19)(18,24)(20,30)(21,25)(22,26)(23,31)(27,32)(28,29)(34,39)(35,42)(41,44)(43,48);(3,12)(6,15)(8,16)(10,13)(17,20,23,22)(18,25,29,27)(19,30,31,26)(21,28,32,24)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,21)(18,26)(19,25)(20,28)(22,24)(23,32)(27,31)(29,30)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,22,23,20)(18,27,29,25)(19,30,31,26)(21,28,32,24)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,19)(18,28)(20,26)(21,25)(22,30)(23,31)(24,29)(27,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=gh,e^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, 8314060);