#10504609

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10411805);