#9455661

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72219217, #72219283
maximal quotient of	#72219216

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9362857);