#8750755

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	#36675	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8657951);