#9938137

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	#49899	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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)(3,8,12,5)(4,15,14,10)(6,16,13,11)(17,18,22,27)(19,24,30,23)(20,25,31,29)(21,28,32,26)(34,39)(35,42)(37,45)(44,47);(1,3,7,12)(2,8,9,5)(4,6,14,13)(10,11,15,16)(17,19,22,30)(18,23,27,24)(20,28,31,26)(21,29,32,25)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(2,9)(3,12)(4,14)(11,16)(17,19,22,30)(18,23,27,24)(20,28,31,26)(21,29,32,25)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,4,7,14)(2,10,9,15)(3,13,12,6)(5,11,8,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,27)(25,32)(26,31)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,5)(2,3)(4,11)(6,15)(7,8)(9,12)(10,13)(14,16)(17,20)(18,25)(19,28)(21,23)(22,31)(24,32)(26,30)(27,29)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,6,7,13)(2,11,9,16)(3,14,12,4)(5,10,8,15)(17,21,22,32)(18,26,27,28)(19,29,30,25)(20,24,31,23)(33,37,38,45)(34,41,39,46)(35,43,42,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=gh,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, 9845333);