#10142154

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	#38388	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,23)(19,21,25,31)(20,24,29,32)(26,30,27,28)(33,34,37,40)(35,41,43,38)(36,39,46,44)(42,48,45,47);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(2,3)(7,9)(8,10)(14,15)(18,23)(19,25)(24,32)(26,27)(33,35,37,43)(34,38,40,41)(36,42,46,45)(39,47,44,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,35)(34,38)(36,45)(37,43)(39,48)(40,41)(42,46)(44,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,20,22,29)(18,24,23,32)(19,26,25,27)(21,30,31,28)(33,36)(34,39)(35,42)(37,46)(38,47)(40,44)(41,48)(43,45);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,19)(20,28)(22,31)(23,25)(24,27)(26,32)(29,30)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,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, 10049350);