#9706717

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	#44253	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4241 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9613913);