#8687349

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4088 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8594545);