#9915335

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

4555 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)(3,10)(4,9)(5,12)(6,8)(7,14)(11,16)(13,15)(17,18)(19,25)(20,24)(21,26)(22,23)(27,28)(29,31)(30,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,3)(2,7)(4,12)(5,13)(6,14)(8,10)(9,16)(11,15)(17,18)(19,21)(20,24)(22,23)(25,26)(27,30)(28,32)(29,31)(35,41)(38,45)(42,47)(46,48);(2,8)(4,15)(7,10)(11,12)(18,23)(19,26)(24,31)(27,32)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,4)(2,9)(3,12)(5,10)(6,15)(7,16)(8,13)(11,14)(17,19)(18,21)(20,27)(22,26)(23,25)(24,30)(28,31)(29,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3)(2,10)(4,12)(5,9)(6,14)(7,8)(11,15)(13,16)(17,20,22,29)(18,24,23,31)(19,27,26,32)(21,30,25,28)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,5)(2,11)(3,13)(4,10)(6,16)(7,15)(8,12)(9,14)(17,21)(18,19)(20,28)(22,25)(23,26)(24,32)(27,31)(29,30)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)

Polycyclic presentation:

2^9|c^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72219217, #72219279, #72219283
maximal quotient of	#72219216

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9822531);