#10349077

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	#36904	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4130 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,15)(12,16)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30)(33,34)(35,39)(36,43)(37,38)(40,44)(41,48)(42,46)(45,47);(2,7)(3,10)(5,14)(12,15)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31)(34,38)(36,44)(39,46)(41,45);(2,7)(4,8)(9,13)(12,15)(18,20)(23,25)(24,28)(30,31)(33,35,37,42)(34,39,38,46)(36,41,44,45)(40,47,43,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,13)(14,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,36,37,44)(34,40,38,43)(35,41,42,45)(39,47,46,48);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,35)(34,39)(36,45)(37,42)(38,46)(40,48)(41,44)(43,47);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,34)(35,39)(36,40)(37,38)(41,47)(42,46)(43,44)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=gh,e^2=h,f^2=h,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, 10256273);