#10201907

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	#44146	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3942 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18)(19,26)(20,24)(21,29)(22,23)(25,31)(27,28)(30,32)(33,34,39,43)(35,40,37,42)(36,46,38,41)(44,48,45,47);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(18,23)(20,26)(25,29)(28,30)(33,35,39,37)(34,40,43,42)(36,45,38,44)(41,48,46,47);(17,19,22,24)(18,20,23,26)(21,27,31,32)(25,28,29,30)(33,36,39,38)(34,41,43,46)(35,44,37,45)(40,47,42,48);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,30)(31,32)(33,37,39,35)(34,42,43,40)(36,44,38,45)(41,47,46,48);(1,5,6,3)(2,7,9,10)(4,11,13,14)(8,16,12,15)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,31)(29,32)(33,38,39,36)(34,41,43,46)(35,45,37,44)(40,47,42,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,13)(14,16)(17,21,22,31)(18,25,23,29)(19,27,24,32)(20,28,26,30)(33,34)(35,40)(36,41)(37,42)(38,46)(39,43)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=i,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, 10109103);