#10324701

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	#53558	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3966 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)(17,18,22,27)(19,23,28,24)(20,25,30,31)(21,29,32,26)(34,35)(36,38)(41,43)(44,45);(3,11)(5,15)(7,16)(9,12)(17,19)(18,23)(20,29)(21,31)(22,28)(24,27)(25,32)(26,30)(33,34,39,35)(36,42,38,40)(37,41,46,43)(44,48,45,47);(1,3)(2,7)(4,9)(5,13)(6,11)(8,15)(10,16)(12,14)(17,19,22,28)(18,24,27,23)(20,26,30,29)(21,25,32,31)(33,35,39,34)(36,42,38,40)(37,41,46,43)(44,47,45,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,15)(10,16)(12,14)(17,20)(18,25)(19,26)(21,23)(22,30)(24,32)(27,31)(28,29)(33,36,39,38)(34,40,35,42)(37,44,46,45)(41,47,43,48);(1,4)(2,8)(3,12)(5,7)(6,14)(9,11)(10,13)(15,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,27)(29,30)(31,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,9)(3,13)(4,7)(6,15)(8,11)(10,12)(14,16)(17,21)(18,26)(19,25)(20,23)(22,32)(24,30)(27,29)(28,31)(33,38,39,36)(34,42,35,40)(37,45,46,44)(41,48,43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,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, 10231897);