#8925203

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

4146 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gh,e^2=g,f^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, 8832399);