#9814953

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4357 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)(3,8)(4,7)(5,6)(10,15)(11,18)(13,21)(20,23)(25,26,30,28)(27,31,33,34)(29,32,38,36)(35,39,37,40);(2,6)(7,8)(9,10,14,15)(11,12,18,16)(13,17,21,22)(19,20,24,23)(27,33)(31,34)(35,37)(39,40);(9,11,14,18)(10,16,15,12)(13,19,21,24)(17,23,22,20)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,40)(37,38);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,24)(22,23)(25,28,30,26)(27,34,33,31)(29,36,38,32)(35,40,37,39);(1,2)(3,7)(4,8)(5,6)(9,12)(10,11)(13,20)(14,16)(15,18)(17,19)(21,23)(22,24)(25,27)(26,31)(28,34)(29,37)(30,33)(32,40)(35,38)(36,39);(1,4,5,3)(2,7,6,8)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gi,f^2=gh,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, 9722149);