#8728256

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

3840 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,22,26)(19,27,28,23)(20,24,30,32)(21,25,29,31)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,22,28)(18,23,26,27)(20,29,30,21)(24,25,32,31)(35,41)(38,45)(42,47)(46,48);(3,10)(6,7)(11,16)(14,15)(17,19,22,28)(18,23,26,27)(20,29,30,21)(24,25,32,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,22,30)(18,24,26,32)(19,21,28,29)(23,31,27,25)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,18)(19,23)(20,24)(21,31)(22,26)(25,29)(27,28)(30,32)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21)(18,25)(19,20)(22,29)(23,32)(24,27)(26,31)(28,30)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8635452);