#9910777

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	#51142	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=h,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, 9817973);