#9909991

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	#53628	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(19,25)(23,29)(26,31)(30,32)(33,34,39,37)(35,42,36,40)(38,41,46,45)(43,48,44,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(18,22)(23,29)(24,27)(30,32)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(2,7)(6,13)(8,11)(14,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31)(33,37,39,34)(35,40,36,42)(38,41,46,45)(43,48,44,47);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,38,39,46)(34,41,37,45)(35,43,36,44)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817187);