#10202065

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4109 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,34,40,44)(35,43,39,38)(36,45,47,41)(37,46,48,42);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31)(33,35,40,39)(34,38,44,43)(36,42,47,46)(37,45,48,41);(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(33,36,40,47)(34,41,44,45)(35,42,39,46)(37,43,48,38);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31)(33,37,40,48)(34,42,44,46)(35,45,39,41)(36,38,47,43);(1,5,6,3)(2,7,9,10)(4,11,13,14)(8,16,12,15)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31)(33,38)(34,35)(36,37)(39,44)(40,43)(41,42)(45,46)(47,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,13)(14,16)(17,20)(18,23)(19,26)(21,30)(22,31)(24,28)(25,32)(27,29)(33,39,40,35)(34,43,44,38)(36,46,47,42)(37,41,48,45)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10109261);