#8932723

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

4129 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)(4,9)(5,16)(6,7)(8,12)(10,13)(14,15)(19,25)(20,28)(23,26)(24,29)(34,39)(40,43)(41,46)(47,48);(1,3,7,12)(2,8,6,11)(4,13,15,5)(9,16,14,10)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,31)(30,32)(33,34,38,39)(35,40,36,43)(37,41,45,46)(42,47,44,48);(2,6)(8,11)(9,14)(10,16)(18,22)(23,26)(24,29)(31,32)(33,35,38,36)(34,40,39,43)(37,44,45,42)(41,48,46,47);(1,4,7,15)(2,9,6,14)(3,5,12,13)(8,10,11,16)(17,19)(18,23)(20,30)(21,25)(22,26)(24,32)(27,28)(29,31)(33,36,38,35)(34,40,39,43)(37,44,45,42)(41,47,46,48);(1,5)(2,10)(3,4)(6,16)(7,13)(8,9)(11,14)(12,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32)(33,34)(35,40)(36,43)(37,41)(38,39)(42,47)(44,48)(45,46);(1,6,7,2)(3,11,12,8)(4,14,15,9)(5,16,13,10)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,37,38,45)(34,41,39,46)(35,42,36,44)(40,47,43,48)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=g,f^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, 8839919);