#8165176

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	#57271	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=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, 8072372);