#8097723

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4070 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004919);