#9850629

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	#50135	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,e^2=gi,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, 9757825);