#10481014

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3848 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=i,e^2=g,f^2=g,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, 10388210);