#8324417

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	#52632	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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)(3,6,9,15)(4,5,8,14)(11,12,13,16)(17,18,23,26)(19,25,21,24)(20,29,30,22)(27,28,31,32)(33,34,39,38)(35,43,45,40)(36,46,44,41)(37,47,48,42);(1,3)(2,6)(4,11)(5,12)(7,9)(8,13)(10,15)(14,16)(17,19,23,21)(18,24,26,25)(20,27,30,31)(22,28,29,32)(33,35,39,45)(34,40,38,43)(36,37,44,48)(41,47,46,42);(3,9)(6,15)(11,13)(12,16)(17,19,23,21)(18,25,26,24)(20,27,30,31)(22,32,29,28)(33,36,39,44)(34,41,38,46)(35,37,45,48)(40,47,43,42);(1,4)(2,5)(3,11)(6,12)(7,8)(9,13)(10,14)(15,16)(17,20)(18,22)(19,27)(21,31)(23,30)(24,28)(25,32)(26,29)(33,35)(34,40)(36,37)(38,43)(39,45)(41,47)(42,46)(44,48);(1,5)(2,8)(3,12)(4,10)(6,13)(7,14)(9,16)(11,15)(17,21,23,19)(18,24,26,25)(20,27,30,31)(22,32,29,28)(33,37)(34,42)(35,44)(36,45)(38,47)(39,48)(40,41)(43,46);(1,6,7,15)(2,9,10,3)(4,12,8,16)(5,13,14,11)(17,22)(18,20)(19,28)(21,32)(23,29)(24,27)(25,31)(26,30)(33,38,39,34)(35,43,45,40)(36,41,44,46)(37,47,48,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8231613);