#10202035

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

3875 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)(4,13)(5,12)(6,8)(7,14)(9,15)(11,16)(17,18)(19,22)(20,25)(21,26)(23,24)(27,32)(28,30)(29,31)(33,34,39,37)(35,43,46,40)(36,47,45,41)(38,42,44,48);(1,3,6,14)(2,7,8,10)(4,12,15,11)(5,13,16,9)(17,19,23,27)(18,22,24,32)(20,28,29,21)(25,30,31,26)(33,35)(34,40)(36,44)(37,43)(38,45)(39,46)(41,48)(42,47);(2,8)(4,15)(7,10)(11,12)(18,24)(19,27)(20,29)(26,30);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,20)(18,25)(19,28)(21,27)(22,30)(23,29)(24,31)(26,32)(33,36,39,45)(34,41,37,47)(35,44,46,38)(40,48,43,42);(1,3)(2,10)(4,11)(5,9)(6,14)(7,8)(12,15)(13,16)(17,21,23,28)(18,26,24,30)(19,29,27,20)(22,31,32,25)(33,37,39,34)(35,43,46,40)(36,41,45,47)(38,42,44,48);(1,5)(2,11)(3,13)(4,7)(6,16)(8,12)(9,14)(10,15)(17,22)(18,27)(19,24)(20,30)(21,25)(23,32)(26,29)(28,31)(33,38,39,44)(34,42,37,48)(35,45,46,36)(40,47,43,41)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=ghi,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, 10109231);