#7812053

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	#40585	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4730 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)(9,10,15,16)(11,18,12,13)(14,17,22,23)(19,24,20,21)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,13,16,18)(14,20,22,19)(17,24,23,21)(25,27)(26,31)(28,29)(30,36)(32,40)(33,38)(34,35)(37,39);(10,16)(11,12)(14,22)(21,24)(27,36)(29,39)(31,35)(33,40);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,13,16,18)(14,20,22,19)(17,21,23,24)(25,28,30,37)(26,32,34,38)(27,29,36,39)(31,40,35,33);(9,13)(10,12)(11,16)(14,21)(15,18)(17,20)(19,23)(22,24)(25,26)(27,31)(28,38)(29,33)(30,34)(32,37)(35,36)(39,40);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,29)(26,33)(27,37)(28,36)(30,39)(31,32)(34,40)(35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,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, 7719249);