#9910993

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3810 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=h,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, 9818189);