#8095360

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(4,5)(6,7)(9,13)(10,14)(11,12)(15,16)(17,18,23,27)(19,28,21,24)(20,26,22,25)(29,31,30,32)(37,45)(40,44)(42,48)(46,47);(3,10)(8,14)(11,15)(12,16)(17,19,23,21)(18,24,27,28)(20,30,22,29)(25,32,26,31)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(2,7)(4,9)(8,14)(11,15)(17,20,23,22)(18,25,27,26)(19,29,21,30)(24,31,28,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3,6,10)(2,8,7,14)(4,11,9,15)(5,12,13,16)(17,18)(19,28)(20,25)(21,24)(22,26)(23,27)(29,32)(30,31);(1,4,6,9)(2,5,7,13)(3,11,10,15)(8,12,14,16)(17,21,23,19)(18,24,27,28)(20,29,22,30)(25,32,26,31)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(1,5)(2,9)(3,12)(4,7)(6,13)(8,15)(10,16)(11,14)(17,22,23,20)(18,26,27,25)(19,29,21,30)(24,31,28,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

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

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, 8002556);