#10168367

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	#43417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4369 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=i,e^2=ghi,f^2=h,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, 10075563);