#10444906

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	#55236	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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:

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

Polycyclic presentation:

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

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, 10352102);