#8125451

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8032647);