#7734355

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	C₂×#534	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4616 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)(9,10,14,19)(11,17,20,15)(12,16,22,23)(13,18,24,21)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,21)(13,23)(14,20)(16,24)(17,19)(18,22)(25,26,30,31)(27,28,32,35)(29,37,38,33)(34,40,39,36);(11,20)(12,22)(15,17)(16,23)(26,31)(28,35)(33,37)(36,40);(1,3,4,7)(2,6,5,8)(9,12,14,22)(10,16,19,23)(11,21,20,18)(13,15,24,17)(25,27,30,32)(26,28,31,35)(29,34,38,39)(33,36,37,40);(9,11)(10,17)(12,21)(13,16)(14,20)(15,19)(18,22)(23,24)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,37)(38,40);(9,13)(10,18)(11,16)(12,15)(14,24)(17,22)(19,21)(20,23)(25,29)(26,33)(27,34)(28,36)(30,38)(31,37)(32,39)(35,40)

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,f^b=fi,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, 7641551);