#8107649

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014845);