#8728379

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4084 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,25,26,21)(20,28,30,24)(27,31,32,29)(33,34)(35,44)(36,37)(38,42)(39,43)(40,45)(41,47)(46,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,18,22,23)(19,25,26,21)(20,24,30,28)(27,29,32,31)(33,35,39,45)(34,40,43,44)(36,46,41,38)(37,42,47,48);(3,10)(6,7)(11,16)(14,15)(18,23)(19,26)(24,28)(27,32)(33,35,39,45)(34,40,43,44)(36,46,41,38)(37,42,47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,19,22,26)(18,21,23,25)(20,27,30,32)(24,31,28,29)(33,36,39,41)(34,37,43,47)(35,38,45,46)(40,48,44,42);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,20,22,30)(18,24,23,28)(19,27,26,32)(21,31,25,29)(33,37)(34,41)(35,42)(36,43)(38,40)(39,47)(44,46)(45,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21)(18,19)(20,29)(22,25)(23,26)(24,32)(27,28)(30,31)(33,38)(34,42)(35,36)(37,44)(39,46)(40,47)(41,45)(43,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8635575);