#9937779

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	#56614	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4122 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)(4,10)(5,13)(6,8)(9,12)(11,16)(14,15)(17,18,22,24)(19,20,25,27)(21,23,30,32)(26,28,29,31)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,3,6,12)(2,7,8,9)(4,13,15,11)(5,14,16,10)(19,25)(20,27)(26,29)(28,31)(34,40)(35,37)(42,46)(43,45);(2,8)(3,12)(10,14)(11,13)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,30)(31,32)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,3)(2,9)(4,13)(5,14)(6,12)(7,8)(10,16)(11,15)(17,20)(18,19)(21,28)(22,27)(23,26)(24,25)(29,32)(30,31)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,4)(2,10)(3,13)(5,7)(6,15)(8,14)(9,16)(11,12)(17,19)(18,20)(21,29)(22,25)(23,31)(24,27)(26,30)(28,32)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,5,6,16)(2,11,8,13)(3,14,12,10)(4,9,15,7)(17,21)(18,23)(19,26)(20,28)(22,30)(24,32)(25,29)(27,31)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=h,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, 9844975);