#8754952

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

3765 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8662148);