#8094599

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	#53106	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8001795);