#9910131

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	#50011	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4168 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,25)(19,21,27,23)(20,24,28,31)(26,29,32,30)(33,34,39,42)(35,40,36,37)(38,41,47,48)(43,46,44,45);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(20,28)(24,31)(26,32)(29,30)(33,35,39,36)(34,37,42,40)(38,43,47,44)(41,45,48,46);(2,3)(7,9)(8,10)(14,15)(17,19,22,27)(18,23,25,21)(20,26,28,32)(24,30,31,29)(33,35,39,36)(34,40,42,37)(38,43,47,44)(41,46,48,45);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,32)(30,31)(33,36,39,35)(34,37,42,40)(38,44,47,43)(41,45,48,46);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,37)(34,35)(36,42)(38,46)(39,40)(41,44)(43,48)(45,47);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,19)(20,29)(22,23)(24,26)(25,27)(28,30)(31,32)(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=gh,d^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, 9817327);