#10412106

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 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)(3,9)(4,11)(5,8)(6,7)(10,16)(12,13)(14,15)(17,18,23,27)(19,26,30,24)(20,22,21,25)(28,29,31,32)(33,34,39,42)(35,40,46,48)(36,41,44,45)(37,47,43,38);(2,7)(4,9)(8,15)(12,16)(17,19)(18,24)(20,31)(21,28)(22,29)(23,30)(25,32)(26,27)(33,35)(34,40)(36,37)(38,45)(39,46)(41,47)(42,48)(43,44);(17,20,23,21)(18,25,27,22)(19,28,30,31)(24,29,26,32)(33,36)(34,41)(35,43)(37,46)(38,40)(39,44)(42,45)(47,48);(1,3,6,11)(2,4,7,9)(5,13,14,10)(8,16,15,12)(17,21,23,20)(18,22,27,25)(19,28,30,31)(24,29,26,32)(33,36)(34,41)(35,37)(38,48)(39,44)(40,47)(42,45)(43,46);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,19)(18,26)(20,28)(21,31)(22,29)(23,30)(24,27)(25,32)(33,37)(34,38)(35,44)(36,46)(39,43)(40,41)(42,47)(45,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,22)(18,20)(19,29)(21,27)(23,25)(24,31)(26,28)(30,32)(33,38)(34,37)(35,45)(36,40)(39,47)(41,46)(42,43)(44,48)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,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, 10319302);