#6343307

order	512
nilpotency class	3
isoclinism	isoclinic class of #17547
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62816	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,e^2=h,f^2=i,b^a=bf,f^a=fh,f^b=fi,g^b=ghi,d^c=dh

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250503);