#10188277

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

3855 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)(3,12)(4,14)(5,11)(6,8)(7,9)(10,15)(13,16)(17,18,22,25)(19,23,29,32)(20,24,30,21)(26,28,27,31)(33,34,39,38)(35,42,45,40)(36,41,37,46)(43,48,44,47);(1,3,7,6)(2,8,9,12)(4,13,15,11)(5,10,16,14)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,32)(26,30)(33,35)(34,40)(36,43)(37,44)(38,42)(39,45)(41,47)(46,48);(2,9)(4,15)(8,12)(11,13)(17,20)(18,24)(19,26)(21,25)(22,30)(23,28)(27,29)(31,32)(33,36,39,37)(34,41,38,46)(35,43,45,44)(40,47,42,48);(1,4)(2,10)(3,13)(5,12)(6,11)(7,15)(8,16)(9,14)(17,20)(18,24)(19,27)(21,25)(22,30)(23,31)(26,29)(28,32)(33,37,39,36)(34,41,38,46)(35,44,45,43)(40,47,42,48);(1,5,7,16)(2,11,9,13)(3,14,6,10)(4,8,15,12)(17,19)(18,23)(20,26)(21,31)(22,29)(24,28)(25,32)(27,30)(33,38,39,34)(35,42,45,40)(36,46,37,41)(43,48,44,47);(1,6,7,3)(2,12,9,8)(4,11,15,13)(5,14,16,10)(17,21,22,24)(18,20,25,30)(19,28,29,31)(23,27,32,26)(33,35)(34,42)(36,43)(37,44)(38,40)(39,45)(41,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=gh,e^2=g,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, 10095473);