#8125475

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	#52904	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3911 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=i,e^2=g,f^2=ghi,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, 8032671);