#9716163

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	#2828	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3809 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=i,f^2=ghi,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, 9623359);