#8994308

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	#42555	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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)(3,10)(4,7)(5,8)(6,9)(11,13)(12,15)(14,16)(33,34,38,42)(35,43,44,39)(36,40,46,47)(37,41,45,48);(1,3)(2,4)(5,11)(6,7)(8,12)(9,10)(13,16)(14,15)(17,18,22,26)(19,24,29,23)(20,21,25,32)(27,31,30,28)(35,44)(37,45)(39,43)(41,48);(1,4,6,10)(2,7,9,3)(5,13,14,12)(8,11,16,15)(17,19,22,29)(18,23,26,24)(20,30,25,27)(21,31,32,28)(33,35,38,44)(34,39,42,43)(36,37,46,45)(40,48,47,41);(1,4)(2,3)(5,13)(6,10)(7,9)(8,15)(11,16)(12,14)(17,20,22,25)(18,21,26,32)(19,27,29,30)(23,28,24,31)(33,36,38,46)(34,40,42,47)(35,45,44,37)(39,41,43,48);(1,5,6,14)(2,8,9,16)(3,11,7,15)(4,12,10,13)(17,19)(18,24)(20,30)(21,28)(22,29)(23,26)(25,27)(31,32)(33,37,38,45)(34,41,42,48)(35,36,44,46)(39,47,43,40);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,21)(18,25)(19,28)(20,26)(22,32)(23,30)(24,27)(29,31)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8901504);