#8180364

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	#46731	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4071 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087560);