#9173179

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3869 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:

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

Polycyclic presentation:

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

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, 9080375);