#7981465

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	#42296	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4421 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7888661);