#8720472

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

3962 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)(4,14)(5,16)(7,11)(8,13)(9,15)(10,12)(17,18,23,27)(19,28,30,24)(20,25,29,21)(22,26,31,32)(33,34,40,44)(35,39,37,42)(36,43,38,41)(45,48,46,47);(1,3)(2,8)(4,5)(6,11)(7,13)(9,16)(10,14)(12,15)(17,19,23,30)(18,24,27,28)(20,22,29,31)(21,26,25,32)(33,35)(34,39)(36,46)(37,40)(38,45)(41,48)(42,44)(43,47);(1,4,7,15)(2,9,11,14)(3,5,13,12)(6,10,8,16)(17,20,23,29)(18,25,27,21)(19,22,30,31)(24,32,28,26)(33,36,40,38)(34,41,44,43)(35,45,37,46)(39,47,42,48);(1,4)(2,9)(3,12)(5,13)(6,16)(7,15)(8,10)(11,14)(17,20)(18,21)(19,22)(23,29)(24,26)(25,27)(28,32)(30,31)(33,37,40,35)(34,42,44,39)(36,45,38,46)(41,47,43,48);(1,5,7,12)(2,10,11,16)(3,4,13,15)(6,9,8,14)(17,21)(18,20)(19,26)(22,24)(23,25)(27,29)(28,31)(30,32)(33,38,40,36)(34,43,44,41)(35,45,37,46)(39,47,42,48);(1,6,7,8)(2,3,11,13)(4,10,15,16)(5,14,12,9)(17,22)(18,26)(19,29)(20,30)(21,28)(23,31)(24,25)(27,32)(33,39,40,42)(34,37,44,35)(36,47,38,48)(41,46,43,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627668);