#10037714

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	#55915	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9944910);