#420048

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327244);