#8925243

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

4200 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=h,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, 8832439);