#8581770

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	#57477	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=i,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488966);