#8108729

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	#40221	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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)(3,5,10,8)(4,12,14,9)(11,15,16,13)(17,18)(19,25)(20,21)(22,31)(23,24)(26,30)(27,28)(29,32)(36,43)(40,45)(42,47)(46,48);(2,7)(3,10)(9,12)(11,16)(18,24)(21,26)(25,32)(27,31)(34,38)(35,41)(40,45)(42,47);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,19,23,29)(18,25,24,32)(20,28,30,22)(21,27,26,31)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(17,20)(18,26)(19,28)(21,24)(22,29)(23,30)(25,31)(27,32);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,21,23,26)(18,20,24,30)(19,27,29,31)(22,25,28,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,22,23,28)(18,27,24,31)(19,20,29,30)(21,25,26,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015925);