#9937715

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3871 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)(3,5)(4,13)(6,15)(7,8)(9,14)(10,12)(11,16)(17,18,22,24)(19,20,25,27)(21,23,30,32)(26,28,29,31)(33,34)(35,36)(37,40)(38,46)(39,42)(41,47)(43,48)(44,45);(1,3)(2,5)(4,6)(7,11)(8,16)(9,10)(12,14)(13,15)(19,25)(20,27)(26,29)(28,31)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,48,46,44);(2,8)(3,11)(4,14)(10,15)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,30)(31,32)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,48,46,44);(1,4)(2,9)(3,12)(5,15)(6,11)(7,14)(8,13)(10,16)(17,20)(18,19)(21,28)(22,27)(23,26)(24,25)(29,32)(30,31)(33,36,39,40)(34,35,42,37)(38,44,47,48)(41,43,46,45);(1,5)(2,3)(4,15)(6,13)(7,16)(8,11)(9,12)(10,14)(17,19)(18,20)(21,29)(22,25)(23,31)(24,27)(26,30)(28,32)(33,37,39,35)(34,40,42,36)(38,43,47,45)(41,44,46,48);(1,6,7,12)(2,10,8,15)(3,4,11,14)(5,9,16,13)(17,21)(18,23)(19,26)(20,28)(22,30)(24,32)(25,29)(27,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=g,e^2=g,f^2=gh,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, 9844911);