#8810094

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	#36332	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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)(17,18,23,21)(19,24,29,31)(20,26,30,25)(22,32,28,27);(2,7)(5,15)(8,11)(9,13)(17,19,23,29)(18,24,21,31)(20,22,30,28)(25,27,26,32)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(3,12)(4,14)(8,11)(9,13)(17,20,23,30)(18,25,21,26)(19,28,29,22)(24,32,31,27)(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)(17,20)(18,26)(19,22)(21,25)(23,30)(24,32)(27,31)(28,29)(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)(17,21,23,18)(19,24,29,31)(20,25,30,26)(22,32,28,27)(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)(17,22)(18,27)(19,20)(21,32)(23,28)(24,25)(26,31)(29,30)(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=g,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, 8717290);