#8119302

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	#57609	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8026498);