#10439973

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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)(3,4)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(17,18)(19,23)(20,28)(21,25)(22,27)(24,29)(26,30)(31,32)(33,34,39,44)(35,42,37,40)(36,41,43,38)(45,46,47,48);(1,2,6,10)(3,7,8,4)(5,9,14,16)(11,15,13,12)(35,37)(40,42)(45,47)(46,48);(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35,39,37)(34,40,44,42)(36,45,43,47)(38,46,41,48);(1,3)(2,7)(4,10)(5,13)(6,8)(9,12)(11,14)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,36)(34,41)(35,45)(37,47)(38,44)(39,43)(40,48)(42,46);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,13,16,11)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^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, 10347169);