#8417136

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3769 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324332);