#7826672

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	#42997	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4698 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,e^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7733868);