#10171866

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=ghi,e^2=gi,f^2=g,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, 10079062);