#8709279

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3911 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8616475);