#7641606

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4330 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=i,e^2=i,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, 7548802);