#9478347

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

4088 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9385543);