#10324622

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	#40740	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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)(8,15)(9,12)(10,16)(11,14)(17,18)(19,27)(20,29)(21,26)(22,24)(23,30)(25,31)(28,32)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(1,3,6,9)(2,4,7,12)(5,14,15,10)(8,16,13,11)(17,19,22,30)(18,23,24,27)(20,26,31,28)(21,29,32,25)(35,43)(37,47)(39,48)(41,44);(2,7)(3,9)(8,13)(10,14)(18,24)(21,32)(23,27)(25,29)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,14)(15,16)(17,20,22,31)(18,25,24,29)(19,28,30,26)(21,23,32,27)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,21,22,32)(18,26,24,28)(19,29,30,25)(20,23,31,27)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,5)(2,8)(3,10)(4,11)(6,15)(7,13)(9,14)(12,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=g,e^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, 10231818);