#8127639

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	#57474	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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(19,25)(22,28)(26,31)(29,32)(34,39)(35,42)(41,44)(43,48);(2,5)(4,14)(8,10)(11,13)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,34,38,39)(35,40,42,37)(36,41,46,44)(43,47,48,45);(1,4)(2,9)(3,11)(5,12)(6,13)(7,14)(8,15)(10,16)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,5,7,2)(3,10,6,8)(4,12,14,9)(11,16,13,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,6,7,3)(2,10,5,8)(4,13,14,11)(9,16,12,15)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034835);