#8720633

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

3880 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)(3,10)(4,13)(5,15)(6,8)(7,9)(11,16)(12,14)(17,18,23,24)(19,21,20,25)(22,26,30,31)(27,29,28,32)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(1,3,7,6)(2,8,9,10)(4,14,15,11)(5,12,13,16)(18,24)(19,20)(26,31)(27,28)(35,41)(38,44)(42,47)(45,48);(2,9)(4,15)(8,10)(11,14)(17,19,23,20)(18,25,24,21)(22,27,30,28)(26,32,31,29)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,4)(2,5)(3,11)(6,14)(7,15)(8,16)(9,13)(10,12)(17,20,23,19)(18,21,24,25)(22,28,30,27)(26,29,31,32)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,5)(2,4)(3,12)(6,16)(7,13)(8,14)(9,15)(10,11)(17,21)(18,19)(20,24)(22,29)(23,25)(26,27)(28,31)(30,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,6,7,3)(2,8,9,10)(4,11,15,14)(5,12,13,16)(17,22,23,30)(18,26,24,31)(19,27,20,28)(21,29,25,32)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=i,d^2=i,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, 8627829);