#9909830

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	#45529	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4164 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)(17,18,23,25)(19,26,21,24)(20,22,29,32)(27,31,30,28)(33,34)(35,40)(36,45)(37,42)(38,46)(39,41)(43,48)(44,47);(3,9)(7,13)(10,15)(14,16)(17,19,23,21)(18,24,25,26)(20,27,29,30)(22,28,32,31)(33,35)(34,40)(36,37)(38,44)(39,43)(41,48)(42,45)(46,47);(2,6)(7,13)(8,11)(14,16)(18,25)(20,29)(24,26)(27,30)(34,41)(35,43)(36,46)(42,44);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,22)(19,27)(21,30)(23,29)(24,28)(25,32)(26,31)(33,36)(34,38)(35,37)(39,46)(40,44)(41,45)(42,48)(43,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21,23,19)(18,24,25,26)(20,30,29,27)(22,28,32,31)(33,37)(34,42)(35,36)(38,48)(39,47)(40,45)(41,44)(43,46);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22)(18,20)(19,28)(21,31)(23,32)(24,27)(25,29)(26,30)(33,38)(34,36)(35,44)(37,40)(39,45)(41,46)(42,43)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^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, 9817026);