#197324

order	512
nilpotency class	3
isoclinism	isoclinic class of #197321
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1213 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104520);