#8925174

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

4310 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)(3,11)(4,8)(5,10)(6,16)(7,9)(12,13)(14,15)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(35,41)(39,46)(42,44)(47,48);(1,3,7,4)(2,8,9,11)(5,12,16,14)(6,13,10,15)(17,19,21,27)(18,22,25,23)(20,26,28,32)(24,29,31,30)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(2,9)(6,10)(8,11)(13,15)(20,28)(24,31)(26,32)(29,30)(34,38)(39,46)(40,43)(47,48);(1,4,7,3)(2,8,9,11)(5,12,16,14)(6,15,10,13)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,5)(2,10)(3,12)(4,14)(6,9)(7,16)(8,15)(11,13)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,6,7,10)(2,5,9,16)(3,13,4,15)(8,12,11,14)(17,20,21,28)(18,24,25,31)(19,26,27,32)(22,29,23,30)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)

Polycyclic presentation:

2^9|b^2=g,d^2=h,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, 8832370);