#10195772

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

3833 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)(3,8)(4,6)(5,12)(7,11)(9,14)(10,15)(13,16)(18,23)(19,26)(25,32)(27,28)(33,34,38,41)(35,42,37,39)(36,40,45,48)(43,47,44,46);(1,3,6,11)(2,7,4,8)(5,10,14,13)(9,15,12,16)(17,18,22,23)(19,24,26,21)(20,25,30,32)(27,29,28,31)(35,37)(39,42)(43,44)(46,47);(2,4)(7,8)(9,12)(15,16)(33,35,38,37)(34,39,41,42)(36,44,45,43)(40,47,48,46);(1,4,6,2)(3,8,11,7)(5,12,14,9)(10,16,13,15)(17,19)(18,24)(20,28)(21,23)(22,26)(25,31)(27,30)(29,32)(33,36,38,45)(34,40,41,48)(35,43,37,44)(39,46,42,47);(1,3)(2,8)(4,7)(5,13)(6,11)(9,15)(10,14)(12,16)(17,20,22,30)(18,25,23,32)(19,27,26,28)(21,31,24,29)(33,37,38,35)(34,39,41,42)(36,43,45,44)(40,47,48,46);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,28,32,27)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=h,e^2=gh,f^2=ghi,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, 10102968);