#9851247

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	#48661	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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)(3,9)(4,13)(5,14)(6,8)(7,12)(10,11)(15,16)(17,18)(19,26)(20,28)(21,24)(22,25)(23,30)(27,29)(31,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,3)(2,7)(4,10)(5,11)(6,12)(8,9)(13,15)(14,16)(17,19,22,24)(18,21,25,26)(20,27,30,31)(23,29,28,32)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,8)(4,14)(7,9)(10,16)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,4)(2,5)(3,10)(6,14)(7,11)(8,13)(9,15)(12,16)(17,20)(18,23)(19,27)(21,29)(22,30)(24,31)(25,28)(26,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,3)(2,9)(4,10)(5,15)(6,12)(7,8)(11,13)(14,16)(17,21,22,26)(18,24,25,19)(20,29,30,32)(23,31,28,27)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,5)(2,4)(3,11)(6,13)(7,10)(8,14)(9,16)(12,15)(17,18)(19,21)(20,23)(22,25)(24,26)(27,29)(28,30)(31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,e^2=ghi,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, 9758443);