#7640894

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4422 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)(11,13)(12,20)(14,17)(15,18)(16,22)(19,23)(21,24)(25,26,30,29)(27,31,34,38)(28,32,37,36)(33,39,35,40);(2,5)(6,8)(9,11,15,17)(10,13,18,14)(12,19,22,24)(16,21,20,23)(25,27,30,34)(26,31,29,38)(28,35,37,33)(32,40,36,39);(9,12)(10,16)(11,19)(13,21)(14,23)(15,22)(17,24)(18,20)(25,28)(26,32)(27,33)(29,36)(30,37)(31,39)(34,35)(38,40);(9,13,15,14)(10,17,18,11)(12,21,22,23)(16,24,20,19)(25,29,30,26)(27,31,34,38)(28,36,37,32)(33,39,35,40);(9,14,15,13)(10,17,18,11)(12,21,22,23)(16,19,20,24)(28,37)(32,36)(33,35)(39,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

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, 7548090);