#8096380

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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)(3,5)(4,9)(6,7)(8,10)(11,14)(12,16)(13,15)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31);(2,7)(3,10)(9,15)(11,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(34,38)(39,45)(40,46)(47,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,12)(13,16)(14,15)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32);(1,4)(2,9)(3,11)(5,14)(6,13)(7,15)(8,12)(10,16)(17,20,22,19)(18,24,26,23)(21,27,29,28)(25,30,32,31)(33,35,37,42)(34,39,38,45)(36,43,44,41)(40,48,46,47);(1,5,6,8)(2,3,7,10)(4,12,13,14)(9,16,15,11)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,36,37,44)(34,40,38,46)(35,41,42,43)(39,47,45,48)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,e^2=gi,f^2=i,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, 8003576);