#7640824

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	#42177	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4380 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:

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

Polycyclic presentation:

2^9|a^2=h,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, 7548020);