#8522159

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	#52765	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3992 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8429355);