#10422685

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=h,e^2=gh,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, 10329881);