#8496491

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	#42533	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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:

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

Polycyclic presentation:

2^9|c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8403687);