#8897999

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	#46640	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8805195);