#10171828

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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)(17,18,22,23)(19,20,25,24)(21,29,31,26)(27,30,32,28)(33,34)(35,39)(36,40)(37,48)(38,42)(41,45)(43,44)(46,47);(1,3)(2,8)(4,12)(5,11)(6,13)(7,14)(9,16)(10,15)(18,23)(20,24)(26,29)(28,30)(33,35,38,44)(34,39,42,43)(36,45,47,37)(40,41,46,48);(17,19,22,25)(18,24,23,20)(21,27,31,32)(26,30,29,28)(33,34,38,42)(35,43,44,39)(36,40,47,46)(37,41,45,48);(1,4,7,13)(2,9,5,15)(3,12,14,6)(8,16,11,10)(17,19)(18,24)(20,23)(21,27)(22,25)(26,30)(28,29)(31,32)(33,34)(35,39)(36,46)(37,41)(38,42)(40,47)(43,44)(45,48);(1,5,7,2)(3,11,14,8)(4,9,13,15)(6,10,12,16)(17,20,22,24)(18,25,23,19)(21,30,31,28)(26,27,29,32)(33,36,38,47)(34,40,42,46)(35,37,44,45)(39,48,43,41);(1,6,7,12)(2,10,5,16)(3,13,14,4)(8,15,11,9)(17,21,22,31)(18,26,23,29)(19,27,25,32)(20,28,24,30)(33,37)(34,41)(35,36)(38,45)(39,46)(40,43)(42,48)(44,47)

Polycyclic presentation:

2^9|a^2=g,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, 10079024);