#10504488

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

4112 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=h,e^2=ghi,f^2=gh,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, 10411684);