#8678968

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

3869 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8586164);