#8324526

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	#58364	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=gh,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8231722);