#8925306

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
prime radical	#36923	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|b^2=gh,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, 8832502);