#8810210

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	#57389	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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)(3,11,12,8)(4,9,14,13)(5,10,15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31);(2,7)(5,15)(8,11)(9,13)(18,21)(20,28)(23,25)(26,29)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(3,12)(4,14)(8,11)(9,13)(17,19,22,27)(18,23,21,25)(20,29,28,26)(24,32,30,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,13,15,9)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,29)(30,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,11,15,8)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,5)(2,10)(3,13)(4,11)(6,15)(7,16)(8,14)(9,12)(17,21,22,18)(19,25,27,23)(20,30,28,24)(26,32,29,31)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8717406);