#10366161

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

3965 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10273357);