#9910999

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,e^2=h,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, 9818195);