#8932780

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

4114 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:

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

Polycyclic presentation:

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

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, 8839976);