#10142155

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	#57210	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4102 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10049351);