#9716345

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	#39144	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4139 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)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31)(33,34,39,41)(35,43,38,40)(36,42,46,37)(44,48,47,45);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31)(33,35,39,38)(34,40,41,43)(36,44,46,47)(37,48,42,45);(2,7)(6,13)(8,11)(14,15)(18,23)(19,26)(24,29)(27,32)(34,41)(36,46)(40,43)(44,47);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,36)(34,42)(35,44)(37,41)(38,47)(39,46)(40,45)(43,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,37,39,42)(34,36,41,46)(35,45,38,48)(40,47,43,44);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,38,39,35)(34,43,41,40)(36,47,46,44)(37,45,42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9623541);