#8118624

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	#52481	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=ghi,e^2=g,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, 8025820);