#8926982

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	#46458	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^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, 8834178);