#9814638

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	#49177	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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:

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

Polycyclic presentation:

2^9|a^2=ghi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9721834);