#7647104

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×#540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4305 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)(9,10,14,18)(11,17,13,15)(12,16,22,20)(19,24,21,23)(26,32)(30,34)(33,36)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24)(25,26)(27,36)(28,33)(29,34)(30,35)(31,32)(37,38)(39,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,22)(21,24)(25,27,31,28)(26,33,32,36)(29,37,35,39)(30,38,34,40);(9,10)(11,15)(12,20)(13,17)(14,18)(16,22)(19,24)(21,23)(25,28,31,27)(26,33,32,36)(29,37,35,39)(30,40,34,38);(9,12,14,22)(10,16,18,20)(11,19,13,21)(15,23,17,24)(25,29,31,35)(26,34,32,30)(27,37,28,39)(33,40,36,38);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,17,18,15)(12,21,22,19)(16,24,20,23)(25,30,31,34)(26,35,32,29)(27,38,28,40)(33,39,36,37)

Polycyclic presentation:

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

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, 7554300);