#8158958

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	#42234	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=g,f^2=gh,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, 8066154);