#10504474

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

3858 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=ghi,f^2=h,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, 10411670);