#9909736

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	#57737	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4171 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9816932);