#8479555

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3838 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)(3,9)(4,13)(5,16)(6,10)(7,8)(11,15)(12,14)(20,28)(23,29)(27,31)(30,32)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(2,8)(5,10)(9,13)(11,14)(17,18)(19,26)(20,29)(21,24)(22,25)(23,28)(27,30)(31,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3,7,4)(2,9,8,13)(5,11,10,14)(6,12,16,15)(17,19,22,24)(18,21,25,26)(20,27,28,31)(23,30,29,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,4,7,3)(2,9,8,13)(5,14,10,11)(6,12,16,15)(17,18)(19,21)(20,23)(22,25)(24,26)(27,30)(28,29)(31,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,5,7,10)(2,6,8,16)(3,11,4,14)(9,12,13,15)(17,20)(18,23)(19,27)(21,30)(22,28)(24,31)(25,29)(26,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47);(1,6)(2,10)(3,12)(4,15)(5,8)(7,16)(9,14)(11,13)(17,21)(18,24)(19,25)(20,30)(22,26)(23,31)(27,29)(28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386751);