#8930179

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3773 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)(20,29)(21,22)(23,26)(25,32)(27,31)(28,30)(34,40)(35,45)(36,38)(43,48);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19,23,20)(18,24,26,29)(21,30,25,27)(22,28,32,31)(33,34)(35,46)(36,42)(37,48)(38,44)(39,40)(41,45)(43,47);(2,6)(3,9)(8,11)(10,12)(17,20,23,19)(18,24,26,29)(21,30,25,27)(22,31,32,28)(33,35)(34,41)(36,37)(38,47)(39,45)(40,46)(42,43)(44,48);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(33,36,39,38)(34,42,40,44)(35,37,45,47)(41,43,46,48);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,21,23,25)(18,22,26,32)(19,27,20,30)(24,31,29,28)(33,37,39,47)(34,43,40,48)(35,36,45,38)(41,42,46,44);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,22)(18,25)(19,28)(20,31)(21,26)(23,32)(24,27)(29,30)(33,38,39,36)(34,44,40,42)(35,47,45,37)(41,48,46,43)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=i,e^2=h,f^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, 8837375);