#9851349

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,e^2=hi,f^2=g,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, 9758545);