#7745327

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	#41404	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4580 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)(11,18)(12,13)(14,17)(15,16)(19,24)(20,21)(22,23)(25,26,30,34)(27,33,35,31)(28,29,32,39)(36,37,40,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,13,16,18)(14,19,22,20)(17,21,23,24)(25,27)(26,31)(28,36)(29,38)(30,35)(32,40)(33,34)(37,39);(10,16)(11,12)(14,22)(21,24)(27,35)(31,33)(36,40)(37,38);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,13,16,18)(14,20,22,19)(17,21,23,24)(25,28,30,32)(26,29,34,39)(27,36,35,40)(31,38,33,37);(9,13)(10,11)(12,16)(14,21)(15,18)(17,19)(20,23)(22,24)(25,29)(26,32)(27,37)(28,34)(30,39)(31,36)(33,40)(35,38);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,27)(26,33)(28,36)(29,37)(30,35)(31,34)(32,40)(38,39)

Polycyclic presentation:

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

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, 7652523);