#8268592

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	#54465	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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:

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

Polycyclic presentation:

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

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, 8175788);