#9935878

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	#42532	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4222 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)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(3,9)(7,13)(10,15)(14,16)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(2,6)(7,13)(8,11)(14,16)(18,25)(19,20)(26,32)(27,28);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9843074);