#8933194

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	#48505	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840390);