#7707034

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4414 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,19)(13,22)(14,17)(16,20)(18,24)(21,23)(25,26,30,29)(27,31,34,37)(28,32,36,38)(33,39,35,40);(2,5)(6,8)(9,11,14,13)(10,15,17,22)(12,18,20,21)(16,23,19,24)(25,27,30,34)(26,31,29,37)(28,35,36,33)(32,40,38,39);(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,19)(22,24)(25,28)(26,32)(27,33)(29,38)(30,36)(31,39)(34,35)(37,40);(12,20)(16,19)(18,21)(23,24)(25,29,30,26)(27,31,34,37)(28,32,36,38)(33,40,35,39);(9,13,14,11)(10,15,17,22)(12,21,20,18)(16,23,19,24)(28,36)(32,38)(33,35)(39,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7614230);