#8757289

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

3973 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)(3,11)(4,13)(5,15)(6,7)(8,12)(9,14)(10,16)(17,18,23,28)(19,24,31,30)(20,27,22,25)(21,26,29,32)(33,34,39,37)(35,36,42,40)(38,45,46,41)(43,47,48,44);(2,7)(5,16)(8,11)(9,13)(17,19)(18,24)(20,21)(22,29)(23,31)(25,32)(26,27)(28,30)(34,37)(35,42)(41,45)(43,48);(2,7)(3,12)(9,13)(10,15)(17,20,23,22)(18,25,28,27)(19,29,31,21)(24,26,30,32)(33,35,39,42)(34,40,37,36)(38,43,46,48)(41,47,45,44);(1,3)(2,8)(4,10)(5,13)(6,12)(7,11)(9,16)(14,15)(17,21,23,29)(18,26,28,32)(19,20,31,22)(24,27,30,25)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,48)(46,47);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,16)(12,15)(17,22,23,20)(18,27,28,25)(19,29,31,21)(24,26,30,32)(33,37,39,34)(35,36,42,40)(38,45,46,41)(43,44,48,47);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,18)(19,30)(20,25)(21,26)(22,27)(23,28)(24,31)(29,32)(33,38,39,46)(34,41,37,45)(35,43,42,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664485);