#8324217

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	#50979	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 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)(19,25)(22,28)(26,31)(29,32)(33,34)(35,44)(36,38)(37,46)(39,43)(40,47)(41,48)(42,45);(1,3)(2,6)(4,11)(5,12)(7,9)(8,13)(10,15)(14,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,35)(34,40)(36,45)(37,48)(38,46)(39,47)(41,42)(43,44);(3,9)(6,15)(11,13)(12,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36,39,48)(34,41,43,38)(35,45,47,37)(40,42,44,46);(1,4)(2,5)(3,11)(6,12)(7,8)(9,13)(10,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,37)(34,42)(35,36)(38,40)(39,45)(41,44)(43,46)(47,48);(1,5)(2,8)(3,12)(4,10)(6,13)(7,14)(9,16)(11,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,35)(34,40)(36,45)(37,48)(38,46)(39,47)(41,42)(43,44);(1,6,7,15)(2,9,10,3)(4,12,8,16)(5,13,14,11)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,38,39,41)(34,36,43,48)(35,46,47,42)(37,40,45,44)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,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, 8231413);