#9909823

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	#2679	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3989 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|c^2=g,d^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, 9817019);