#7770298

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	#40118	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4300 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7677494);