#8924964

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

4290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8832160);