#8761946

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4034 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8669142);