#7658211

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 7565407);