#9848609

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	#37788	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4277 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,e^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, 9755805);