#8750216

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	#36696	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=gh,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, 8657412);