#8750764

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	#36677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=i,e^2=gh,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, 8657960);