#8119901

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	#47176	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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)(3,4)(5,12)(6,14)(7,8)(9,15)(10,16)(11,13)(17,18,23,24)(19,21,20,25)(22,26,30,31)(27,29,28,32)(35,41)(38,44)(42,47)(45,48);(1,3,7,5)(2,4,8,12)(6,10,15,13)(9,11,14,16)(18,24)(19,20)(26,31)(27,28)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(2,8)(3,5)(9,14)(10,13)(17,19,23,20)(18,25,24,21)(22,27,30,28)(26,32,31,29)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,4)(2,3)(5,8)(6,11)(7,12)(9,10)(13,14)(15,16)(17,20,23,19)(18,21,24,25)(22,28,30,27)(26,29,31,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,5,7,3)(2,4,8,12)(6,13,15,10)(9,11,14,16)(17,21)(18,19)(20,24)(22,29)(23,25)(26,27)(28,31)(30,32)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,6,7,15)(2,9,8,14)(3,10,5,13)(4,11,12,16)(17,22,23,30)(18,26,24,31)(19,27,20,28)(21,29,25,32)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8027097);