#10168474

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	#50354	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=i,e^2=ghi,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, 10075670);