#7640916

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	#41306	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4391 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=gi,e^2=g,f^2=h,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, 7548112);