#10166169

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	#49700	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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)(3,9,10,4)(5,13,14,8)(11,12,16,15)(18,22)(19,25)(24,27)(26,28)(33,34,40,42)(35,43,39,41)(36,38,37,47)(44,48,45,46);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35,40,39)(34,41,42,43)(36,44,37,45)(38,46,47,48);(2,7)(3,10)(8,13)(11,16)(17,19,21,25)(18,23,22,30)(20,26,29,28)(24,31,27,32)(33,36,40,37)(34,38,42,47)(35,44,39,45)(41,46,43,48);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,37,40,36)(34,38,42,47)(35,45,39,44)(41,46,43,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,20,21,29)(18,24,22,27)(19,26,25,28)(23,31,30,32)(33,38)(34,37)(35,46)(36,42)(39,48)(40,47)(41,45)(43,44);(1,5)(2,8)(3,11)(4,12)(6,14)(7,13)(9,15)(10,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,39,40,35)(34,41,42,43)(36,45,37,44)(38,46,47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=gh,e^2=hi,f^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, 10073365);