#8930503

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	#46310	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4470 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8837699);