#10504989

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	#48176	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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)(3,8)(4,14)(5,9)(6,15)(7,11)(10,12)(13,16)(17,18)(19,20)(21,29)(22,23)(24,27)(25,31)(26,32)(28,30)(33,34,38,41)(35,37,40,43)(36,39,46,48)(42,45,44,47);(1,3,7,13)(2,8,11,16)(4,12,5,6)(9,15,14,10)(18,23)(20,24)(25,29)(28,32)(33,35)(34,37)(36,44)(38,40)(39,47)(41,43)(42,46)(45,48);(1,4,7,5)(2,9,11,14)(3,12,13,6)(8,15,16,10)(17,19)(18,24)(20,23)(21,26)(22,27)(25,32)(28,29)(30,31)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,48)(43,47);(1,5,7,4)(2,9,11,14)(3,6,13,12)(8,15,16,10)(17,20,22,24)(18,19,23,27)(21,28,31,32)(25,26,29,30)(33,34)(35,43)(36,39)(37,40)(38,41)(42,47)(44,45)(46,48);(1,2)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,19)(18,20)(21,30)(22,27)(23,24)(25,32)(26,31)(28,29)(33,37,38,43)(34,40,41,35)(36,45,46,47)(39,44,48,42);(1,6)(2,10)(3,4)(5,13)(7,12)(8,9)(11,15)(14,16)(17,21)(18,25)(19,26)(20,28)(22,31)(23,29)(24,32)(27,30)(33,36)(34,39)(35,44)(37,47)(38,46)(40,42)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10412185);