#10325773

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	#56638	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=g,e^2=gh,f^2=g,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, 10232969);