#8888155

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	#47621	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4171 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18)(19,25)(20,28)(21,24)(22,26)(23,30)(27,31)(29,32)(35,41)(38,42)(43,47)(46,48);(2,7)(4,8)(5,14)(11,16)(17,18,21,24)(19,22,26,25)(20,23,30,28)(27,31,29,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,42)(43,48)(46,47);(2,7)(3,9)(10,13)(11,16)(19,26)(22,25)(27,29)(31,32)(33,34,37,40)(35,38,41,42)(36,39,45,44)(43,46,47,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19)(18,22)(20,29)(21,26)(23,32)(24,25)(27,30)(28,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,45)(46,47);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20)(18,23)(19,27)(21,30)(22,31)(24,28)(25,32)(26,29)(33,35,37,41)(34,38,40,42)(36,43,45,47)(39,46,44,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,31)(28,30)(29,32)(33,36,37,45)(34,39,40,44)(35,43,41,47)(38,46,42,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8795351);